- Email: [email protected]
Structural and electronic inhomogeneity for graphene grown on the C-face of SiC: Insights from ab initio calculations
Accepted Manuscript Title: Structural and electronic inhomogeneity for graphene grown on the C-face of SiC: insights from ab initio calculations Author: I. Deretzis A. La Magna PII: DOI: Reference:
S0169-4332(13)01767-4 http://dx.doi.org/doi:10.1016/j.apsusc.2013.09.126 APSUSC 26409
To appear in:
APSUSC
Received date: Revised date: Accepted date:
14-6-2013 20-9-2013 21-9-2013
Please cite this article as: I. Deretzis, A. La Magna, Structural and electronic inhomogeneity for graphene grown on the C-face of SiC: insights from ab initio calculations, Applied Surface Science (2013), http://dx.doi.org/10.1016/j.apsusc.2013.09.126 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Ac
ce pt
ed
M
an
us
cr
ip t
DFT calculations for epitaxial graphene that grows on unreconstructed and reconstructed SiC(0001) surfaces Structural and electronic inhomogeinity for graphene grown on different types of SiC(000-1) surfaces
Page 1 of 13
*Manuscript
cr
I. Deretzis, A. La Magna
ip t
Structural and electronic inhomogeneity for graphene grown on the C-face of SiC: insights from ab initio calculations
an
us
Istituto per la Microelettronica e Microsistemi (CNR-IMM), VIII Strada 5, 95121 Catania, Italy
Abstract
Ac ce p
te
d
M
The graphitization of the SiC(000¯1) plane, commonly referred to as the Cface of SiC, takes place through the sublimation and reorganization of surface atoms upon high-temperature annealing. Often, such reorganization gives rise to ordered atomic reconstructions over the ideally flat (000¯1) plane. In this article, we use the density functional theory to model graphene/SiC(000¯1) interfaces with an (1×1), (2×2) and (3×3) SiC periodicity. Our results indicate that the interface geometry can be crucial for both the stability and the electronic characteristics of the first graphitic layer, revealing a complex scenario of binding, doping and electronic correlations. We argue that the presence of more than one interface geometry at different areas of the same sample could be a reason for structural inhomogeneity and n- to p-type transitions. Keywords: graphene, SiC, C-face, DFT
1
2 3 4 5 6
1. Introduction
One of the fundamental issues for the integration of graphene into devices and applications is the controllable growth of large-scale homogeneous samples with reproducible structural/electronic characteristics. A particular system within this context is epitaxial graphene grown on SiC substrates. Particularity stems from the fact that SiC is not solely a host substrate, Email address: [email protected] (I. Deretzis)
Preprint submitted to Applied Surface Science
September 20, 2013
Page 2 of 13
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
ip t
cr
13
us
12
an
11
M
10
d
9
te
8
but also the growth template for graphene. Graphene can be engineered on SiC surfaces through high-temperature thermal annealing, in conjunction with a precise control over the sublimation rate of Si surface atoms [1, 2]. We note here that such graphene is “ready-to-use” for devices, since by itself SiC is a wide-bandgap semiconductor and no further transfer is necessary. This single-step process however has a significant drawback, since the graphene/SiC interface gets formed during growth and significantly depends on the surface orientation and growth conditions [3, 4, 5, 6, 7, 8]. The most common case is that of graphene grown on the (0001)√plane,√i.e. the Si-face of SiC. Here, the SiC surface reconstructs with a (6 3 × 6 3)R30◦ periodicity [3, 9], forming a C-rich buffer layer with a honeycomb lattice and a covalent bonding to the substrate, over which graphene layers with notable quality grow. However, the “signature” of this interface is visible from the elevated doping level of the first graphene layer (∼1013 cm−2 ) as well as a strong electron-phonon coupling that suppresses the intrinsic mobility of graphene [10]. A more complicated case seems to be that of graphene grown on the C-face of SiC. Here, contrary to the single-reconstruction of the Si face, both unreconstructed as well as reconstructed surfaces have been reported in the literature [4, 5, 6, 11]. In the case of reconstructed surfaces, various periodicities have been observed (e.g. the (2×2)C [4] and the (3×3) [4, 5]), which often coexist in the same sample. A fundamental question that arises is if such reconstructions can give rise to epitaxial graphenes with different structural and electronic properties. It is known from the experiment that growth of graphene on the C-face is difficult to control and often results in inhomogeneous areas of multilayer structures that seem to be rotationally disordered. The origin of this inhomogeneity is not fully understood to date, and this appears to be a field where theoretical calculations can be helpful. ¯ Initial modeling √ of the◦ epitaxial graphene/SiC(0001) system took place √ within the ( 3 × 3)R30 scheme [12, 13] and predicted that a buffer layer similar to that of the Si-face should also exist on the C-fase. Within this model, only the second graphitic layer acquired graphene-like characteristics, with the Fermi level crossing the Dirac point. The presence of a significant mismatch between the two parts of the heterostructure for this model (with either an ∼8% stretching of the graphene lattice or an equal shrinking of the SiC substrate) as well as the lack of experimental evidence on a buffer layer led to a second generation of modeling with unstrained structures, which were constructed using significantly larger supercells [14, 15]. In addition, the observation of reconstructed SiC surfaces at the interface between graphene and
Ac ce p
7
2
Page 3 of 13
65
2. Methodology
53 54 55 56 57 58 59 60 61 62 63
66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
cr
52
us
51
an
50
M
49
d
48
te
47
Epitaxial graphene on the Si-face √ has a unique orientation with respect to √ the SiC substrate, imposed by the (6 3 × 6 3)R30◦ periodicity of the buffer layer. Contrary, for graphene on the C-face multiple orientations have been observed [19]. We therefore model graphene/SiC(000¯1) interfaces with the sole criterion of commensurability, in order to minimize unphysical stresses that do not occur in real systems. In our calculations we consider SiC substrates of the 4H polytype that are modeled with four SiC bilayers passivated by H atoms at the lower termination of the slabs. For the unreconstructed as well as the (2×2)C reconstructed surfaces we use a (4 × 4) SiC supercell that is commensurate with a (5 × 5) graphene cell [15]. In this case, no rotational mismatch exists between the lattice vectors of graphene and the SiC surface. Similarly, for the case of the (3×3) reconstruction we notice that a SiC(6×6) cell has almost the same periodicity with a hexagonal graphene cell which is rotated by an angle φ = 6.6◦ with respect to a zigzag edge [18]. Formally, such graphene supercell can be defined by a lattice vector r=7a1 +a2 , where
Ac ce p
46
ip t
64
SiC(000¯1) led to the development of models where the graphene layer relaxes on Si/C adatoms rather than the bare SiC surface [16, 17, 18]. A striking issue that emerged from such simulations was the variability of the results depending on the adopted model and the choice of the exchange-correlation functional. Such variability made difficult a direct comparative evaluation of the results for the cases of the various reconstructed and unreconstructed surfaces. In this article, we critically affront this issue using a common computational framework (same computational code, same exchange-correlation functionals, same pseudopotentials) and the density functional theory (DFT). We study the structural and electronic characteristics of graphene grown on the C-face of SiC for various interface geometries. Our calculations show that the formation of graphene/SiC interface can be fundamental for the electronic dispersion, the doping level and the structural stability of the first graphene layer that grows on the (000¯1) surface. We also observe p to n-type transitions when passing from unreconstructed to reconstructed surfaces. The article is organized as follows: In Sec. 2 we introduce the computational methodology. In Sec. 3 we study the case of graphene on the unreconstructed SiC(000¯1) surface. In Sec 4 we perform calculations for graphene on the (2×2)C (Subsec. 4.1) and the (3×3) (Subsec. 4.2) reconstructions. Finally, in Sec. 5 we discuss our results.
45
3
Page 4 of 13
91
3. Unreconstructed SiC(000¯ 1) surface
89
92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116
cr
88
us
87
an
86
The simplest modeling scheme for the study of the epitaxial graphene/SiC(000¯1) heterosystem is to consider an ideal (000¯1) plane below a graphene layer [15]. Experimentally, this approach is supported by low-energy electron diffraction measurements obtained for samples grown at very high temperatures, where only (1×1) patterns are clearly visible [11]. As a first configuration we consider a single graphene layer that relaxes on the unreconstructed C-plane [Fig. 1(a)], using the LDA as the exchange-correlation functional. Upon kinetic equilibrium, graphene remains almost flat with a ∼ 0.13 ˚ A thickness and an average distance of 2.82 ˚ A from the surface C atoms. However, it is important to notice here the structural relaxation of the SiC surface below the graphene layer: C surface atoms are subjected to a strong bulking, reducing the C-Si surface mean bond-length by ∼ 3.2% with respect to the bulk value. Similarly, the mean bond-angle distribution centered at the surface atoms acquires a value of ∼ 116◦ , which is higher than the 109.5◦ value of the bulk, but still smaller than the 120◦ value of the pure sp2 hybridization. Such structural information implies that in unreconstructed graphene/SiC(000¯1) interfaces, the C surface atoms tend to lose their sp3 hybridization and acquire an sp2 one. However, from an electronic viewpoint, “fingerprints” of both hybridizations are present in the electronic band structure [Fig. 1(b)]. The sp3 character is visible from the dispersion of a large number of quasi-flat bands around the Fermi level of the heterosystem, which originate from the danging bonds of the C surface atoms. Interestingly, the sp2 character of the surface C atoms is visible from the modification of the π bands of graphene, which loose their linearity around the Dirac point. In this case, a peculiar π-type bilayer system is formed, where the upper layer is the graphene sheet
M
85
d
84
te
83
Ac ce p
82
ip t
90
a1 and a2 are the basis vectors of the hexagonal Bravais lattice of graphene [18]. The DFT calculations are performed with the SIESTA code [20]. Electronic correlations are treated within the Local Density Approximation (LDA) [21]. However, in particular cases we compare these results with respective ones obtained within the Generalized Gradient Approximation (GGA) [22]. The core electron contributions are described with standard norm-conserving nonlocal pseudopotentials [23]. The Brillouin zone sampling takes place within a Monkhorst-Pack scheme [24], setting the k-grid cutoff parameter to 16 ˚ A. All atoms are allowed to relax until forces are less than 0.04 eV/˚ A.
81
4
Page 5 of 13
(b)
GGA
LDA
0.5
ip t
E - EF (eV)
1
0 -0.5 -1 Γ
Μ
Κ
cr
(a)
Γ
Γ
Μ
Κ
Γ
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141
M
120
d
119
and the lower layer is a hexagonal sublattice composed of the surface atoms of the SiC substrate. In this configuration, the graphene sheet is p-doped, with a carrier concentration of ∼1012 cm−2 . We now repeat the same calculation using the Perdew-Burke-Ernzerhof functional [22] of the GGA. The most important difference with respect to the LDA is that the graphene layer has a distance of 3.27 ˚ A from the SiC surface. Such difference reflects the absence of a proper treatment of van der Waals interactions in both the LDA and the GGA. This structural divergence also has implications in the electronic spectrum of the heterosystem, where the π-bands of the graphene layer have a form that is closer to the typical linear dispersion of freestanding graphene, although not perfectly linear also here. In both cases, the graphene layer maintains a metallic character and a p-type doping. In order to further investigate possible stable configurations other than the planar one we have repeated the previous calculations by imposing a smaller distance between graphene and the SiC substrate as an initial condition. We found that the graphene layer partially binds to the substrate, forming covalent bonds with a small percentage of the C surface atoms [Fig. 2(a)]. Such atoms loose their sp2 character for the sp3 one, whereas all the rest (almost 4/5ths of the graphene atoms and the corresponding surface atoms below them) maintain electronic characteristics that are similar to the planar configuration. This coupling inhomogeneity gives rise to a significant corrugation for the graphitic layer (with a thickness of ∼ 1 ˚ A for the LDA), which now looks more like a “buffer” layer similar to the one of the graphene/SiC(0001) interface. Electronically, the remaining π bands of the
te
118
Ac ce p
117
an
us
Figure 1: (a) Scheme of the unreconstructed graphene/SiC(000¯1) interface. (b) Band structure of the graphene/SiC(000¯1) heterosystem, within the LDA (left) and the GGA (right).
5
Page 6 of 13
(b)
1
ip t
(a)
0
-0.5
us
-1 Γ
cr
E - EF (eV)
0.5
Μ
Κ
Γ
146 147 148 149 150 151 152 153
154
155 156 157 158 159 160 161 162
M
145
d
144
te
143
sp2 -hybridized atoms still disperse throughout the SiC bandgap [Fig. 2(b)], without maintaining though any similarity with the linear dispersion of the Dirac bands. From an energetic viewpoint, the LDA shows that such configuration has a lower energy with respect to the planar one, even though differences are very small (only 23 meV per graphene atom). On the contrary, the planar configuration remains the most stable for the GGA (for 44 meV per graphene atom) due to an excess of the deformation energy provoked by a stronger rippling of the graphitic layer. The previous discrepancies indicate that a proper treatment of the nonlocal dispersive interactions within the DFT could be enlightening for this system. However, due to the marginal differences, an atomic scale experimental investigation similar to the one of Ref. 25 is necessary.
Ac ce p
142
an
Figure 2: (a) Scheme of a bound configuration for the unreconstructed graphene/SiC(000¯1) interface. (b) Band structure of the graphene/SiC(000¯1) heterostructure shown in (a) within the LDA.
4. Reconstructed SiC(000¯ 1) surfaces Even if unreconstructed graphene/SiC(000¯1) interfaces are plausible, experimental techniques that probe for the structural characteristics of the graphene/SiC(000¯1) interface with atomic resolution have shown that the SiC surface either reconstructs in an orderly manner [4] or that at least it shows structural divergence from the ideal SiC bilayer structure [6]. In the following subparagraphs we study the structural and electronic properties for graphene on the (2×2)C and the (3×3) reconstructed surfaces, for which atomic models exist in the literature [26, 18].
6
Page 7 of 13
ip t
0.5
0
-0.5
us
-1
cr
(b) E - EF (eV)
(a)
-1.5
Γ
Μ
Κ
Γ
166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185
M
d
165
te
164
4.1. Graphene on the (2×2)C SiC(000¯1) surface The (2×2)C -reconstructed graphene/SiC(000¯1) interface was first observed by Hiebel et al. [4] through scanning tunneling microscopy (STM) measurements, and thanks to the model of Seubert et al. [26] it has been also theoretically investigated within the DFT [16, 17]. Here we repeat such calculations under the computational scheme used in the present study, in order to have an equivalent comparative framework with the other reconstructed and non-reconstructed SiC surfaces. We start by the description of the model which contains a single Si adatom per SiC(2×2) surface cell. In the lowest energy configuration, such adatom relaxes at the H3 position of the surface [Fig. 3(a)] and saturates three C dangling bonds. In this geometry, there are two remaining dangling bonds in the supercell, associated with the Si adatom and a C restatom. Our results confirm that graphene does not covalently bind to this surface and maintains its intrinsic low-energy electronic spectrum [Fig. 3(b)], in accordance with previous studies. However, there are some quantitative aspects that diverse and are worth commenting. Calculations using the LDA show that graphene should be highly n-doped (∼ 1013 cm−2 ). This result is qualitatively in line with a previous study using both LDA and van der Waals functionals [17] but in disagreement with calculations based on the GGA [16, 17]. Moreover, it shows a higher doping, which is due to the relatively small distance between the graphene layer and the reconstructed surface (< d >= 2.42 ˚ A here), enhancing the consequent charge transfer from the substrate towards graphene (which is fundamentally
Ac ce p
163
an
Figure 3: (a) Scheme of the (2×2)C -reconstructed graphene/SiC(000¯1) interface. (b) “Fat band” representation of the band structure for the orbitals of graphene. Calculations are based on the LDA.
7
Page 8 of 13
193
194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
ip t
cr
192
us
191
4.2. Graphene on the (3×3) SiC(000¯1) surface Similarly to the (2×2)C case, the (3×3)-reconstructed graphene/SiC(000¯1) interface was observed through STM [4], whereas the structural and electronic properties of the bare SiC(3×3) reconstruction were known since the work of Hoster et al. [27]. Recently [18], we proposed a atomic scheme that computationally reproduces the STM spectra and morphological model of Ref. 27. It can summarized as follows: each bare SiC(3×3) surface cell contains four Si adatoms (in an inverse tetrahedral configuration with respect to the surface) and six C adatoms that stabilize this tetrahedral structure. The whole ad-structure lies in two levels in terms of height (with the first level being occupied by C atoms and a single Si atom while the second being occupied by three Si atoms). The presence of dangling bonds is limited in the three upper Si atoms of the tetrahedron and in two C restatoms of the bare surface. However, due to the vertical thickness of the reconstruction, only the upper three dangling bonds can be resolved in the STM spectra [27]. The DFT-LDA results for the heterosystem [Fig. 4(a)] show that the (3×3)reconstruction should be the less interacting with graphene with respect to the previously studied cases. Indeed, the average separation of the interface is 3.1 ˚ A, the binding energy is minimal (see Table 1) and doping is almost absent (∼ 1011 cm−2 ). The graphene layer is quasi-flat with a peak-to-valley thickness of 0.13 ˚ A and the typical Dirac cone stays almost intact (with a sub-kB T perturbation at the Dirac point for room temperatures) [Fig. 4(b)]. However, the intrinsic metallicity of polar SiC surfaces is reflected in the resonance of the C dangling bond states of the SiC surface that disperse around the Dirac point [18]. We finally note that such computational description of the graphene/SiC(3×3) system is in a very good agreement with the experiments.
an
190
M
189
d
188
te
187
a charge transfer from Si to C atoms) and introducing a secondary band gap of ∼ 40 meV at the Dirac point. We note that such levels of doping are rarely observed in the experiments for graphene on the C-face of SiC. The origins of this discrepancy between theory and experiment may be (a) the marginality of the (2×2)C reconstruction in real graphene/SiC(000¯1) samples, (b) the accuracy of the proposed model, and (c) the accuracy of the computational approach. In all cases, a need for a better understanding of the (2×2)C -reconstructed graphene/SiC interface emerges.
Ac ce p
186
8
Page 9 of 13
(b)
E - EF (eV)
0.5
0
us
-0.5
ip t
1
cr
(a)
-1
Μ
Γ
Κ
224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242
M
Table 1 summarizes the binding energies, structural characteristics and doping levels obtained for the previously reported configurations within the LDA. The first graphitic layer that grows on the C-face of SiC has graphenelike characteristics only in the case of reconstructed surfaces, whereas in the case of the unreconstructed surface it partially maintains its π bands and metallic character, but not the linear dispersion around the Dirac point. The less-bound/less-doped graphene is the one that grows on the (3×3)reconstructed surface, whereas questions arise from the high electron doping of the (2×2)C reconstruction which needs further investigation. A transition from p to n-type doping takes place when “switching” from unreconstructed to reconstructed surfaces. Moreover, if we calculate the vertical distance of the first graphitic layer from the bare (000¯1) surface for all configurations (for reconstructed surfaces this distance is equal to < d > plus the thickness of the reconstruction), this varies by ∼ 3 ˚ A. Based on this picture it is interesting to ask which should be the structural/electronic properties of a graphene monolayer in a sample where the previous interfaces coexist. Calculations indicate that areas of different distances from the substrate, variable doping levels and nonuniform electronic characteristics should also coexist. This structural and electronic inhomogeneity could be at the origin of the inhomogeneity often observed for epitaxial graphene grown on the C-face of SiC [11] and may also be related
d
223
te
222
5. Discussion
Ac ce p
221
an
Figure 4: (a) Scheme of the (3×3)-reconstructed graphene/SiC(000¯1) interface. (b) “Fat band” representation of the band structure for the orbitals of graphene. Calculations are based on the LDA.
9
Page 10 of 13
Lr (˚ A) 0.13 0.96 0.23 0.13
doping (cm−2 ) p=6.2×1012 p=2.9×1012 n=4.1×1013 n=4.7×1011
ip t
< d > (˚ A) 2.82 2.29 2.42 3.1
Eb (meV) -110 -322 -115 -54
cr
model (1×1) U (1×1) B (2×2)C (3×3)
an
us
Table 1: Comparison of calculated parameters for the two configurations of the unreconstructed (U: unbound, B: partially bound), the (2×2)C and the (3×3) reconstructed SiC(000¯ 1) surfaces: Average binding energy per graphene atom (Eb ), average distance between graphene and the substrate (< d >), peak-to-valley roughness of the graphene layer (Lr ) and carrier concentration for the graphene layer. Results are obtained within the LDA.
248
References
249 250 251 252
253 254 255
256 257
258 259
260 261
d
246
[1] K. V. Emtsev, A. Bostwick, K. Horn, J. Jobst, G. L. Kellogg, L. Ley, J. L. McChesney, T. Ohta, S. A. Reshanov, J. R¨ohrl, E. Rotenberg, A. K. Schmid, D. Waldmann, H. B. Weber, T. Seyller, Nature Materials 8 (2009) 203–207.
te
245
Ac ce p
244
M
247
to further issues (like the unconventional rotational stacking of multilayer graphene films [19] or the electronic transport properties at the junctions between different reconstructions [28]). We point out that further experimental and theoretical efforts need to concentrate towards the atomic resolution of the graphene/SiC(000¯1) interface.
243
[2] W. A. de Heer, C. Berger, M. Ruan, M. Sprinkle, X. Li, Y. Hu, B. Zhang, J. Hankinson, E. Conrad, Proceedings of the National Academy of Science 108 (2011) 16900–16905. [3] C. Riedl, C. Coletti, U. Starke, Journal of Physics D Applied Physics 43 (2010) K4009. [4] F. Hiebel, P. Mallet, F. Varchon, L. Magaud, J.-Y. Veuillen, Phys. Rev. B 78 (15) (2008) 153412. [5] N. Srivastava, G. He, Luxmi, R. M. Feenstra, Phys. Rev. B 85 (4) (2012) 041404.
10
Page 11 of 13
264
265 266 267
ip t
263
[6] J. Hass, R. Feng, J. E. Mill´an-Otoya, X. Li, M. Sprinkle, P. N. First, W. A. de Heer, E. H. Conrad, C. Berger, Phys. Rev. B 75 (21) (2007) 214109.
[7] A. Ouerghi, A. Balan, C. Castelli, M. Picher, R. Belkhou, M. Eddrief, M. G. Silly, M. Marangolo, A. Shukla, F. Sirotti, Appl. Phys. Lett. 101 (2) (2012) 021603.
cr
262
[8] M. Ostler, I. Deretzis, S. Mammadov, F. Giannazzo, G. Nicotra, C. Spinella, T. Seyller, A. La Magna, Phys. Rev. B 88 (2013) 085408.
270
[9] I. Deretzis, A. La Magna, Phys. Rev. B 84 (23) (2011) 235426.
272
[10] A. J. M. Giesbers, P. Proch´azka, C. F. J. Flipse, Phys. Rev. B 87 (2013) 195405.
an
271
us
269
268
275
[12] A. Mattausch, O. Pankratov, Phys. Rev. Lett. 99 (2007) 076802.
M
274
[11] L. Johansson, S. Watcharinyanon, A. Zakharov, T. Iakimov, R. Yakimova, C. Virojanadara, Phys. Rev. B 84 (12) (2011) 125405.
273
279
[14] O. Pankratov, S. Hensel, M. Bockstedte, Phys. Rev. B 82 (2010) 121416.
280
281 282
283 284
285
286 287 288 289
te
Ac ce p
277
d
278
[13] F. Varchon, R. Feng, J. Hass, X. Li, B. N. Nguyen, C. Naud, P. Mallet, J.-Y. Veuillen, C. Berger, E. H. Conrad, L. Magaud, Phys. Rev. Lett. 99 (2007) 126805.
276
[15] I. Deretzis, A. La Magna, Appl. Phys. Lett. 98 (2) (2011) 023113. [16] L. Magaud, F. Hiebel, F. Varchon, P. Mallet, J.-Y. Veuillen, Phys. Rev. B 79 (2009) 161405. [17] T. Jayasekera, S. Xu, K. Kim, M. Buongiorno-Nardelli, Phys. Rev. B 84 (3) (2011) 035442. [18] I. Deretzis, A. L. Magna, Appl. Phys. Lett. 102 (9) (2013) 093101. [19] M. Sprinkle, J. Hicks, A. Tejeda, A. Taleb-Ibrahimi, P. Le F`evre, F. Bertran, H. Tinkey, M. C. Clark, P. Soukiassian, D. Martinotti, J. Hass, E. H. Conrad, Journal of Physics D Applied Physics 43 (2010) K4006.
11
Page 12 of 13
293
[21] J. P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048–5079.
291
ip t
292
[20] J. M. Soler, E. Artacho, J. D. Gale, A. Garc´ıa, J. Junquera, P. Ordej´on, D. S´anchez-Portal, Journal of Physics Condensed Matter 14 (2002) 2745–2779.
290
296
[23] N. Troullier, J. L. Martins, Phys. Rev. B 43 (1991) 1993–2006.
297
[24] H. J. Monkhorst, J. D. Pack, Phys. Rev. B 13 (1976) 5188–5192.
302 303
304 305
us
an
M
301
[26] A. Seubert, J. Bernhardt, M. Nerding, U. Starke, K. Heinz, Surface Science 454 (2000) 45–48. [27] H. Hoster, M. Kulakov, B. Bullemer, Surface science 382 (1) (1997) L658–L665.
d
300
[28] F. Giannazzo, I. Deretzis, A. La Magna, F. Roccaforte, R. Yakimova, Phys. Rev. B 86 (23) (2012) 235422.
te
299
[25] G. Nicotra, Q. M. Ramasse, I. Deretzis, A. La Magna, C. Spinella, F. Giannazzo, ACS Nano 7 (4) (2013) 3045–3052.
Ac ce p
298
cr
295
[22] J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865– 3868.
294
12
Page 13 of 13
S0169-4332(13)01767-4 http://dx.doi.org/doi:10.1016/j.apsusc.2013.09.126 APSUSC 26409
To appear in:
APSUSC
Received date: Revised date: Accepted date:
14-6-2013 20-9-2013 21-9-2013
Please cite this article as: I. Deretzis, A. La Magna, Structural and electronic inhomogeneity for graphene grown on the C-face of SiC: insights from ab initio calculations, Applied Surface Science (2013), http://dx.doi.org/10.1016/j.apsusc.2013.09.126 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Ac
ce pt
ed
M
an
us
cr
ip t
DFT calculations for epitaxial graphene that grows on unreconstructed and reconstructed SiC(0001) surfaces Structural and electronic inhomogeinity for graphene grown on different types of SiC(000-1) surfaces
Page 1 of 13
*Manuscript
cr
I. Deretzis, A. La Magna
ip t
Structural and electronic inhomogeneity for graphene grown on the C-face of SiC: insights from ab initio calculations
an
us
Istituto per la Microelettronica e Microsistemi (CNR-IMM), VIII Strada 5, 95121 Catania, Italy
Abstract
Ac ce p
te
d
M
The graphitization of the SiC(000¯1) plane, commonly referred to as the Cface of SiC, takes place through the sublimation and reorganization of surface atoms upon high-temperature annealing. Often, such reorganization gives rise to ordered atomic reconstructions over the ideally flat (000¯1) plane. In this article, we use the density functional theory to model graphene/SiC(000¯1) interfaces with an (1×1), (2×2) and (3×3) SiC periodicity. Our results indicate that the interface geometry can be crucial for both the stability and the electronic characteristics of the first graphitic layer, revealing a complex scenario of binding, doping and electronic correlations. We argue that the presence of more than one interface geometry at different areas of the same sample could be a reason for structural inhomogeneity and n- to p-type transitions. Keywords: graphene, SiC, C-face, DFT
1
2 3 4 5 6
1. Introduction
One of the fundamental issues for the integration of graphene into devices and applications is the controllable growth of large-scale homogeneous samples with reproducible structural/electronic characteristics. A particular system within this context is epitaxial graphene grown on SiC substrates. Particularity stems from the fact that SiC is not solely a host substrate, Email address: [email protected] (I. Deretzis)
Preprint submitted to Applied Surface Science
September 20, 2013
Page 2 of 13
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
ip t
cr
13
us
12
an
11
M
10
d
9
te
8
but also the growth template for graphene. Graphene can be engineered on SiC surfaces through high-temperature thermal annealing, in conjunction with a precise control over the sublimation rate of Si surface atoms [1, 2]. We note here that such graphene is “ready-to-use” for devices, since by itself SiC is a wide-bandgap semiconductor and no further transfer is necessary. This single-step process however has a significant drawback, since the graphene/SiC interface gets formed during growth and significantly depends on the surface orientation and growth conditions [3, 4, 5, 6, 7, 8]. The most common case is that of graphene grown on the (0001)√plane,√i.e. the Si-face of SiC. Here, the SiC surface reconstructs with a (6 3 × 6 3)R30◦ periodicity [3, 9], forming a C-rich buffer layer with a honeycomb lattice and a covalent bonding to the substrate, over which graphene layers with notable quality grow. However, the “signature” of this interface is visible from the elevated doping level of the first graphene layer (∼1013 cm−2 ) as well as a strong electron-phonon coupling that suppresses the intrinsic mobility of graphene [10]. A more complicated case seems to be that of graphene grown on the C-face of SiC. Here, contrary to the single-reconstruction of the Si face, both unreconstructed as well as reconstructed surfaces have been reported in the literature [4, 5, 6, 11]. In the case of reconstructed surfaces, various periodicities have been observed (e.g. the (2×2)C [4] and the (3×3) [4, 5]), which often coexist in the same sample. A fundamental question that arises is if such reconstructions can give rise to epitaxial graphenes with different structural and electronic properties. It is known from the experiment that growth of graphene on the C-face is difficult to control and often results in inhomogeneous areas of multilayer structures that seem to be rotationally disordered. The origin of this inhomogeneity is not fully understood to date, and this appears to be a field where theoretical calculations can be helpful. ¯ Initial modeling √ of the◦ epitaxial graphene/SiC(0001) system took place √ within the ( 3 × 3)R30 scheme [12, 13] and predicted that a buffer layer similar to that of the Si-face should also exist on the C-fase. Within this model, only the second graphitic layer acquired graphene-like characteristics, with the Fermi level crossing the Dirac point. The presence of a significant mismatch between the two parts of the heterostructure for this model (with either an ∼8% stretching of the graphene lattice or an equal shrinking of the SiC substrate) as well as the lack of experimental evidence on a buffer layer led to a second generation of modeling with unstrained structures, which were constructed using significantly larger supercells [14, 15]. In addition, the observation of reconstructed SiC surfaces at the interface between graphene and
Ac ce p
7
2
Page 3 of 13
65
2. Methodology
53 54 55 56 57 58 59 60 61 62 63
66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
cr
52
us
51
an
50
M
49
d
48
te
47
Epitaxial graphene on the Si-face √ has a unique orientation with respect to √ the SiC substrate, imposed by the (6 3 × 6 3)R30◦ periodicity of the buffer layer. Contrary, for graphene on the C-face multiple orientations have been observed [19]. We therefore model graphene/SiC(000¯1) interfaces with the sole criterion of commensurability, in order to minimize unphysical stresses that do not occur in real systems. In our calculations we consider SiC substrates of the 4H polytype that are modeled with four SiC bilayers passivated by H atoms at the lower termination of the slabs. For the unreconstructed as well as the (2×2)C reconstructed surfaces we use a (4 × 4) SiC supercell that is commensurate with a (5 × 5) graphene cell [15]. In this case, no rotational mismatch exists between the lattice vectors of graphene and the SiC surface. Similarly, for the case of the (3×3) reconstruction we notice that a SiC(6×6) cell has almost the same periodicity with a hexagonal graphene cell which is rotated by an angle φ = 6.6◦ with respect to a zigzag edge [18]. Formally, such graphene supercell can be defined by a lattice vector r=7a1 +a2 , where
Ac ce p
46
ip t
64
SiC(000¯1) led to the development of models where the graphene layer relaxes on Si/C adatoms rather than the bare SiC surface [16, 17, 18]. A striking issue that emerged from such simulations was the variability of the results depending on the adopted model and the choice of the exchange-correlation functional. Such variability made difficult a direct comparative evaluation of the results for the cases of the various reconstructed and unreconstructed surfaces. In this article, we critically affront this issue using a common computational framework (same computational code, same exchange-correlation functionals, same pseudopotentials) and the density functional theory (DFT). We study the structural and electronic characteristics of graphene grown on the C-face of SiC for various interface geometries. Our calculations show that the formation of graphene/SiC interface can be fundamental for the electronic dispersion, the doping level and the structural stability of the first graphene layer that grows on the (000¯1) surface. We also observe p to n-type transitions when passing from unreconstructed to reconstructed surfaces. The article is organized as follows: In Sec. 2 we introduce the computational methodology. In Sec. 3 we study the case of graphene on the unreconstructed SiC(000¯1) surface. In Sec 4 we perform calculations for graphene on the (2×2)C (Subsec. 4.1) and the (3×3) (Subsec. 4.2) reconstructions. Finally, in Sec. 5 we discuss our results.
45
3
Page 4 of 13
91
3. Unreconstructed SiC(000¯ 1) surface
89
92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116
cr
88
us
87
an
86
The simplest modeling scheme for the study of the epitaxial graphene/SiC(000¯1) heterosystem is to consider an ideal (000¯1) plane below a graphene layer [15]. Experimentally, this approach is supported by low-energy electron diffraction measurements obtained for samples grown at very high temperatures, where only (1×1) patterns are clearly visible [11]. As a first configuration we consider a single graphene layer that relaxes on the unreconstructed C-plane [Fig. 1(a)], using the LDA as the exchange-correlation functional. Upon kinetic equilibrium, graphene remains almost flat with a ∼ 0.13 ˚ A thickness and an average distance of 2.82 ˚ A from the surface C atoms. However, it is important to notice here the structural relaxation of the SiC surface below the graphene layer: C surface atoms are subjected to a strong bulking, reducing the C-Si surface mean bond-length by ∼ 3.2% with respect to the bulk value. Similarly, the mean bond-angle distribution centered at the surface atoms acquires a value of ∼ 116◦ , which is higher than the 109.5◦ value of the bulk, but still smaller than the 120◦ value of the pure sp2 hybridization. Such structural information implies that in unreconstructed graphene/SiC(000¯1) interfaces, the C surface atoms tend to lose their sp3 hybridization and acquire an sp2 one. However, from an electronic viewpoint, “fingerprints” of both hybridizations are present in the electronic band structure [Fig. 1(b)]. The sp3 character is visible from the dispersion of a large number of quasi-flat bands around the Fermi level of the heterosystem, which originate from the danging bonds of the C surface atoms. Interestingly, the sp2 character of the surface C atoms is visible from the modification of the π bands of graphene, which loose their linearity around the Dirac point. In this case, a peculiar π-type bilayer system is formed, where the upper layer is the graphene sheet
M
85
d
84
te
83
Ac ce p
82
ip t
90
a1 and a2 are the basis vectors of the hexagonal Bravais lattice of graphene [18]. The DFT calculations are performed with the SIESTA code [20]. Electronic correlations are treated within the Local Density Approximation (LDA) [21]. However, in particular cases we compare these results with respective ones obtained within the Generalized Gradient Approximation (GGA) [22]. The core electron contributions are described with standard norm-conserving nonlocal pseudopotentials [23]. The Brillouin zone sampling takes place within a Monkhorst-Pack scheme [24], setting the k-grid cutoff parameter to 16 ˚ A. All atoms are allowed to relax until forces are less than 0.04 eV/˚ A.
81
4
Page 5 of 13
(b)
GGA
LDA
0.5
ip t
E - EF (eV)
1
0 -0.5 -1 Γ
Μ
Κ
cr
(a)
Γ
Γ
Μ
Κ
Γ
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141
M
120
d
119
and the lower layer is a hexagonal sublattice composed of the surface atoms of the SiC substrate. In this configuration, the graphene sheet is p-doped, with a carrier concentration of ∼1012 cm−2 . We now repeat the same calculation using the Perdew-Burke-Ernzerhof functional [22] of the GGA. The most important difference with respect to the LDA is that the graphene layer has a distance of 3.27 ˚ A from the SiC surface. Such difference reflects the absence of a proper treatment of van der Waals interactions in both the LDA and the GGA. This structural divergence also has implications in the electronic spectrum of the heterosystem, where the π-bands of the graphene layer have a form that is closer to the typical linear dispersion of freestanding graphene, although not perfectly linear also here. In both cases, the graphene layer maintains a metallic character and a p-type doping. In order to further investigate possible stable configurations other than the planar one we have repeated the previous calculations by imposing a smaller distance between graphene and the SiC substrate as an initial condition. We found that the graphene layer partially binds to the substrate, forming covalent bonds with a small percentage of the C surface atoms [Fig. 2(a)]. Such atoms loose their sp2 character for the sp3 one, whereas all the rest (almost 4/5ths of the graphene atoms and the corresponding surface atoms below them) maintain electronic characteristics that are similar to the planar configuration. This coupling inhomogeneity gives rise to a significant corrugation for the graphitic layer (with a thickness of ∼ 1 ˚ A for the LDA), which now looks more like a “buffer” layer similar to the one of the graphene/SiC(0001) interface. Electronically, the remaining π bands of the
te
118
Ac ce p
117
an
us
Figure 1: (a) Scheme of the unreconstructed graphene/SiC(000¯1) interface. (b) Band structure of the graphene/SiC(000¯1) heterosystem, within the LDA (left) and the GGA (right).
5
Page 6 of 13
(b)
1
ip t
(a)
0
-0.5
us
-1 Γ
cr
E - EF (eV)
0.5
Μ
Κ
Γ
146 147 148 149 150 151 152 153
154
155 156 157 158 159 160 161 162
M
145
d
144
te
143
sp2 -hybridized atoms still disperse throughout the SiC bandgap [Fig. 2(b)], without maintaining though any similarity with the linear dispersion of the Dirac bands. From an energetic viewpoint, the LDA shows that such configuration has a lower energy with respect to the planar one, even though differences are very small (only 23 meV per graphene atom). On the contrary, the planar configuration remains the most stable for the GGA (for 44 meV per graphene atom) due to an excess of the deformation energy provoked by a stronger rippling of the graphitic layer. The previous discrepancies indicate that a proper treatment of the nonlocal dispersive interactions within the DFT could be enlightening for this system. However, due to the marginal differences, an atomic scale experimental investigation similar to the one of Ref. 25 is necessary.
Ac ce p
142
an
Figure 2: (a) Scheme of a bound configuration for the unreconstructed graphene/SiC(000¯1) interface. (b) Band structure of the graphene/SiC(000¯1) heterostructure shown in (a) within the LDA.
4. Reconstructed SiC(000¯ 1) surfaces Even if unreconstructed graphene/SiC(000¯1) interfaces are plausible, experimental techniques that probe for the structural characteristics of the graphene/SiC(000¯1) interface with atomic resolution have shown that the SiC surface either reconstructs in an orderly manner [4] or that at least it shows structural divergence from the ideal SiC bilayer structure [6]. In the following subparagraphs we study the structural and electronic properties for graphene on the (2×2)C and the (3×3) reconstructed surfaces, for which atomic models exist in the literature [26, 18].
6
Page 7 of 13
ip t
0.5
0
-0.5
us
-1
cr
(b) E - EF (eV)
(a)
-1.5
Γ
Μ
Κ
Γ
166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185
M
d
165
te
164
4.1. Graphene on the (2×2)C SiC(000¯1) surface The (2×2)C -reconstructed graphene/SiC(000¯1) interface was first observed by Hiebel et al. [4] through scanning tunneling microscopy (STM) measurements, and thanks to the model of Seubert et al. [26] it has been also theoretically investigated within the DFT [16, 17]. Here we repeat such calculations under the computational scheme used in the present study, in order to have an equivalent comparative framework with the other reconstructed and non-reconstructed SiC surfaces. We start by the description of the model which contains a single Si adatom per SiC(2×2) surface cell. In the lowest energy configuration, such adatom relaxes at the H3 position of the surface [Fig. 3(a)] and saturates three C dangling bonds. In this geometry, there are two remaining dangling bonds in the supercell, associated with the Si adatom and a C restatom. Our results confirm that graphene does not covalently bind to this surface and maintains its intrinsic low-energy electronic spectrum [Fig. 3(b)], in accordance with previous studies. However, there are some quantitative aspects that diverse and are worth commenting. Calculations using the LDA show that graphene should be highly n-doped (∼ 1013 cm−2 ). This result is qualitatively in line with a previous study using both LDA and van der Waals functionals [17] but in disagreement with calculations based on the GGA [16, 17]. Moreover, it shows a higher doping, which is due to the relatively small distance between the graphene layer and the reconstructed surface (< d >= 2.42 ˚ A here), enhancing the consequent charge transfer from the substrate towards graphene (which is fundamentally
Ac ce p
163
an
Figure 3: (a) Scheme of the (2×2)C -reconstructed graphene/SiC(000¯1) interface. (b) “Fat band” representation of the band structure for the orbitals of graphene. Calculations are based on the LDA.
7
Page 8 of 13
193
194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
ip t
cr
192
us
191
4.2. Graphene on the (3×3) SiC(000¯1) surface Similarly to the (2×2)C case, the (3×3)-reconstructed graphene/SiC(000¯1) interface was observed through STM [4], whereas the structural and electronic properties of the bare SiC(3×3) reconstruction were known since the work of Hoster et al. [27]. Recently [18], we proposed a atomic scheme that computationally reproduces the STM spectra and morphological model of Ref. 27. It can summarized as follows: each bare SiC(3×3) surface cell contains four Si adatoms (in an inverse tetrahedral configuration with respect to the surface) and six C adatoms that stabilize this tetrahedral structure. The whole ad-structure lies in two levels in terms of height (with the first level being occupied by C atoms and a single Si atom while the second being occupied by three Si atoms). The presence of dangling bonds is limited in the three upper Si atoms of the tetrahedron and in two C restatoms of the bare surface. However, due to the vertical thickness of the reconstruction, only the upper three dangling bonds can be resolved in the STM spectra [27]. The DFT-LDA results for the heterosystem [Fig. 4(a)] show that the (3×3)reconstruction should be the less interacting with graphene with respect to the previously studied cases. Indeed, the average separation of the interface is 3.1 ˚ A, the binding energy is minimal (see Table 1) and doping is almost absent (∼ 1011 cm−2 ). The graphene layer is quasi-flat with a peak-to-valley thickness of 0.13 ˚ A and the typical Dirac cone stays almost intact (with a sub-kB T perturbation at the Dirac point for room temperatures) [Fig. 4(b)]. However, the intrinsic metallicity of polar SiC surfaces is reflected in the resonance of the C dangling bond states of the SiC surface that disperse around the Dirac point [18]. We finally note that such computational description of the graphene/SiC(3×3) system is in a very good agreement with the experiments.
an
190
M
189
d
188
te
187
a charge transfer from Si to C atoms) and introducing a secondary band gap of ∼ 40 meV at the Dirac point. We note that such levels of doping are rarely observed in the experiments for graphene on the C-face of SiC. The origins of this discrepancy between theory and experiment may be (a) the marginality of the (2×2)C reconstruction in real graphene/SiC(000¯1) samples, (b) the accuracy of the proposed model, and (c) the accuracy of the computational approach. In all cases, a need for a better understanding of the (2×2)C -reconstructed graphene/SiC interface emerges.
Ac ce p
186
8
Page 9 of 13
(b)
E - EF (eV)
0.5
0
us
-0.5
ip t
1
cr
(a)
-1
Μ
Γ
Κ
224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242
M
Table 1 summarizes the binding energies, structural characteristics and doping levels obtained for the previously reported configurations within the LDA. The first graphitic layer that grows on the C-face of SiC has graphenelike characteristics only in the case of reconstructed surfaces, whereas in the case of the unreconstructed surface it partially maintains its π bands and metallic character, but not the linear dispersion around the Dirac point. The less-bound/less-doped graphene is the one that grows on the (3×3)reconstructed surface, whereas questions arise from the high electron doping of the (2×2)C reconstruction which needs further investigation. A transition from p to n-type doping takes place when “switching” from unreconstructed to reconstructed surfaces. Moreover, if we calculate the vertical distance of the first graphitic layer from the bare (000¯1) surface for all configurations (for reconstructed surfaces this distance is equal to < d > plus the thickness of the reconstruction), this varies by ∼ 3 ˚ A. Based on this picture it is interesting to ask which should be the structural/electronic properties of a graphene monolayer in a sample where the previous interfaces coexist. Calculations indicate that areas of different distances from the substrate, variable doping levels and nonuniform electronic characteristics should also coexist. This structural and electronic inhomogeneity could be at the origin of the inhomogeneity often observed for epitaxial graphene grown on the C-face of SiC [11] and may also be related
d
223
te
222
5. Discussion
Ac ce p
221
an
Figure 4: (a) Scheme of the (3×3)-reconstructed graphene/SiC(000¯1) interface. (b) “Fat band” representation of the band structure for the orbitals of graphene. Calculations are based on the LDA.
9
Page 10 of 13
Lr (˚ A) 0.13 0.96 0.23 0.13
doping (cm−2 ) p=6.2×1012 p=2.9×1012 n=4.1×1013 n=4.7×1011
ip t
< d > (˚ A) 2.82 2.29 2.42 3.1
Eb (meV) -110 -322 -115 -54
cr
model (1×1) U (1×1) B (2×2)C (3×3)
an
us
Table 1: Comparison of calculated parameters for the two configurations of the unreconstructed (U: unbound, B: partially bound), the (2×2)C and the (3×3) reconstructed SiC(000¯ 1) surfaces: Average binding energy per graphene atom (Eb ), average distance between graphene and the substrate (< d >), peak-to-valley roughness of the graphene layer (Lr ) and carrier concentration for the graphene layer. Results are obtained within the LDA.
248
References
249 250 251 252
253 254 255
256 257
258 259
260 261
d
246
[1] K. V. Emtsev, A. Bostwick, K. Horn, J. Jobst, G. L. Kellogg, L. Ley, J. L. McChesney, T. Ohta, S. A. Reshanov, J. R¨ohrl, E. Rotenberg, A. K. Schmid, D. Waldmann, H. B. Weber, T. Seyller, Nature Materials 8 (2009) 203–207.
te
245
Ac ce p
244
M
247
to further issues (like the unconventional rotational stacking of multilayer graphene films [19] or the electronic transport properties at the junctions between different reconstructions [28]). We point out that further experimental and theoretical efforts need to concentrate towards the atomic resolution of the graphene/SiC(000¯1) interface.
243
[2] W. A. de Heer, C. Berger, M. Ruan, M. Sprinkle, X. Li, Y. Hu, B. Zhang, J. Hankinson, E. Conrad, Proceedings of the National Academy of Science 108 (2011) 16900–16905. [3] C. Riedl, C. Coletti, U. Starke, Journal of Physics D Applied Physics 43 (2010) K4009. [4] F. Hiebel, P. Mallet, F. Varchon, L. Magaud, J.-Y. Veuillen, Phys. Rev. B 78 (15) (2008) 153412. [5] N. Srivastava, G. He, Luxmi, R. M. Feenstra, Phys. Rev. B 85 (4) (2012) 041404.
10
Page 11 of 13
264
265 266 267
ip t
263
[6] J. Hass, R. Feng, J. E. Mill´an-Otoya, X. Li, M. Sprinkle, P. N. First, W. A. de Heer, E. H. Conrad, C. Berger, Phys. Rev. B 75 (21) (2007) 214109.
[7] A. Ouerghi, A. Balan, C. Castelli, M. Picher, R. Belkhou, M. Eddrief, M. G. Silly, M. Marangolo, A. Shukla, F. Sirotti, Appl. Phys. Lett. 101 (2) (2012) 021603.
cr
262
[8] M. Ostler, I. Deretzis, S. Mammadov, F. Giannazzo, G. Nicotra, C. Spinella, T. Seyller, A. La Magna, Phys. Rev. B 88 (2013) 085408.
270
[9] I. Deretzis, A. La Magna, Phys. Rev. B 84 (23) (2011) 235426.
272
[10] A. J. M. Giesbers, P. Proch´azka, C. F. J. Flipse, Phys. Rev. B 87 (2013) 195405.
an
271
us
269
268
275
[12] A. Mattausch, O. Pankratov, Phys. Rev. Lett. 99 (2007) 076802.
M
274
[11] L. Johansson, S. Watcharinyanon, A. Zakharov, T. Iakimov, R. Yakimova, C. Virojanadara, Phys. Rev. B 84 (12) (2011) 125405.
273
279
[14] O. Pankratov, S. Hensel, M. Bockstedte, Phys. Rev. B 82 (2010) 121416.
280
281 282
283 284
285
286 287 288 289
te
Ac ce p
277
d
278
[13] F. Varchon, R. Feng, J. Hass, X. Li, B. N. Nguyen, C. Naud, P. Mallet, J.-Y. Veuillen, C. Berger, E. H. Conrad, L. Magaud, Phys. Rev. Lett. 99 (2007) 126805.
276
[15] I. Deretzis, A. La Magna, Appl. Phys. Lett. 98 (2) (2011) 023113. [16] L. Magaud, F. Hiebel, F. Varchon, P. Mallet, J.-Y. Veuillen, Phys. Rev. B 79 (2009) 161405. [17] T. Jayasekera, S. Xu, K. Kim, M. Buongiorno-Nardelli, Phys. Rev. B 84 (3) (2011) 035442. [18] I. Deretzis, A. L. Magna, Appl. Phys. Lett. 102 (9) (2013) 093101. [19] M. Sprinkle, J. Hicks, A. Tejeda, A. Taleb-Ibrahimi, P. Le F`evre, F. Bertran, H. Tinkey, M. C. Clark, P. Soukiassian, D. Martinotti, J. Hass, E. H. Conrad, Journal of Physics D Applied Physics 43 (2010) K4006.
11
Page 12 of 13
293
[21] J. P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048–5079.
291
ip t
292
[20] J. M. Soler, E. Artacho, J. D. Gale, A. Garc´ıa, J. Junquera, P. Ordej´on, D. S´anchez-Portal, Journal of Physics Condensed Matter 14 (2002) 2745–2779.
290
296
[23] N. Troullier, J. L. Martins, Phys. Rev. B 43 (1991) 1993–2006.
297
[24] H. J. Monkhorst, J. D. Pack, Phys. Rev. B 13 (1976) 5188–5192.
302 303
304 305
us
an
M
301
[26] A. Seubert, J. Bernhardt, M. Nerding, U. Starke, K. Heinz, Surface Science 454 (2000) 45–48. [27] H. Hoster, M. Kulakov, B. Bullemer, Surface science 382 (1) (1997) L658–L665.
d
300
[28] F. Giannazzo, I. Deretzis, A. La Magna, F. Roccaforte, R. Yakimova, Phys. Rev. B 86 (23) (2012) 235422.
te
299
[25] G. Nicotra, Q. M. Ramasse, I. Deretzis, A. La Magna, C. Spinella, F. Giannazzo, ACS Nano 7 (4) (2013) 3045–3052.
Ac ce p
298
cr
295
[22] J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865– 3868.
294
12
Page 13 of 13