Spontaneous polarization induced doping in quasi-free standing epitaxial graphene on silicon carbide from density functional theory

0.37 MB·English

by  J. Slawinska

Spontaneous polarization induced doping in quasi-free standing epitaxial graphene on
silicon carbide from density functional theory
J. Sl awin´ska,1,2 H. Aramberri,2 and J. I. Cerda´2
1Department of Solid State Physics, University of L ´od´z, Pomorska 149/153, 90236 L ´od´z, Poland
2Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC, Cantoblanco, 28049 Madrid, Spain.
(Dated: January 27, 2015)
By means of density functional theory (DFT) calculations we have quantitatively estimated the
5 impactofthespontaneouspolarization (SP)oftheSiC(0001) substrateontheelectronicproperties
1 of quasi-freestanding graphene (QFG) decoupled from the SiC by H intercalation. To correctly
0 include within standard DFT slab calculations the influence of the SP, which is a bulk property,
2 on a surface confined property such as the graphene’s doping, we attach a double gold layer at the

n C-terminatedbottomoftheslabwhichintroducesametalinducedgapstatethatpinsthechemical
a potential within the gap. Furthermore, expanding the interlayer distances at the bottom of the
J slab creates a local dipole moment which counters that arising from the slab’s polar character and
6 allowstoalignthelocationofthegraphene’sDiracpoint(DP)forcubicSiC(111)withthechemical
2 potential. Thus, the DP shifts obtained for other polytypes under the same slab model become an
almost directmeasurement of theSP-induceddoping. Ourresultsconfirm therecent proposal that

] theSPinduces theexperimentally observed p-typedopingin thegraphenelayer which can achieve
ci DP shifts of up to several hundreds of meV (or equivalently, ∼ 1013e/cm2) for specific polytypes.
s The doping is found to increase with the hexagonality of the polytype and its thickness. For the
- slab thickenssesconsidered (6-12SiCbilayers)an ample,almost continuous,rangeofdopingvalues
l
r can be achieved by tuning the number of stacking defects and their location with respect to the
t
m surface. The slab model is next generalized by performing large scale DFT calculations where self-
dopingisincludedin theQFGviapoint defects(vacancyplusaH atom) thusallowing toestimate
.
t the interplay between both sources of p-doping (SP- versus defect-induced) which turns out to be
a
essentially additive.
m
- PACSnumbers: 73.22.Pr,81.05.ue,77.22.Ej
d
n
o Introduction– Epitaxial graphene (EG) on silicon car- negative SP for all polytypes) as well as its dependence
c
bide (SiC) has been demonstrated to be an excellent on the hexagonality of the polytype, and has found fur-
[
materialforhigh-performancetechnologicalapplications. ther support in several works [11, 18, 19, 23]. Mam-

1 [1–5] On the other hand, quasi-freestanding graphene madov et al [24] have very recently corroborated such
v
(QFG) on SiC obtained from EG by hydrogen interca- model by means of a systematic angle-resolved photoe-
6
8 lation holds an even greater promise for the realization mission electron spectroscopy (ARPES) study for 3C-,
4 of graphene-based electronic devices because of its facile 4H- and 6H-SiC QFG-systems. Two sources of doping
6 large-scale production together with excellent transport are identified in this study: a surface band bending aris-
0
characteristics[6–9]. Thelattermainlyarisefromtheef- ingfromthe bulkdopantsandthe SP.The formerwould
.
1 ficientreductionoftheinteractionbetweengraphene(G) beresponsibleforthemildn-typedopingencounteredin
0 and SiC, otherwise strongly coupled, by the intercalated 3C-SiC(111) samples [10, 18, 24] while the latter would
5
H layer. Although this weak interaction fully preserves account for the 1.5 factor between hole dopings in
1 ∼
the Dirac cones [9–12], a p-type doping is routinely de- 4H- and 6H-SiC samples [24]. Although not addressed
:
v tected in experiments [6, 9, 13–16] which can be as large in that work, a third main source is the self-doping in-
Xi as 5.5 1012 cm−2 as observed by Speck et al [6] for duced by the presence of intrinsic defects in the QFG
QF∼Gsam×pleson6H-SiC(0001)oreven 2.0 1013cm−2 (vacancies and/or adatoms), thoroughly studied within
r
∼ ×
a asrecentlyreportedon4H-SiC(0001)byUrbanetal[17], theory [12, 25] but not so well experimentally [6, 26–29].
while for cubic 3C-SiC(111) Coletti et al found a slight
n-type doping [10, 18]. In this Letter we investigate the relationship between
the SP and the graphene’s p-type doping in perfect as
Basedonthese results,among severalothers[7,8,19– wellas defected QFGfor severalH-passivatedhexagonal
21], Ristein et al [22] pointed to a relationship between SiCpolytypesviadensityfunctionaltheory(DFT)based
the G doping and the hexagonality of the SiC polytype calculations. The importance of employing a first prin-
employedassubstrate,andproposedamacroscopicspon- ciples approach to this end is manifold: it represents a
taneouspolarization(SP)dopingmodelwherebythesub- powerful predictive tool when tuning the surface density
strate’sSPcreatesapseudo-chargeatthesurfaceequiva- ofe/holecarriersinG-baseddevices[1–5,21],providesa
lenttoarealacceptorlayer. Themodelexplainsthesign uniquewaytoquantifyonequalfootingsthecompetition
of the doping (since the Si-terminated surface exhibits a between self-doping and SP and, not the least, allows to
2
explore the validity of the macroscopic dielectric theory placing a gold plane in registry with the carbon atoms
whenthesystemsizeisshrinkedtothenanoscale,asitis andasecondonefollowinganfccstackingsequence(see
not obvious if a direct relationship between the SP and inset in Figure 1(b)). The systems were relaxed freez-
the doping charge in the QFG will still hold. Neverthe- ing the inner SiC BLs to bulk-like positions optimized
less, first principles slab calculations –aiming to model a independently for each polytype. In order to obtain ac-
semi-infinite surface– have not yet been attempted due curatedopingcharges,large(100 100)k-supercellswere
×
to at least three non-trivial issues which need to be re- employed to sample the Brillouin zone [35].
solved: (i)combiningabulkandasurfaceconfinedprop-
erty (SP and G’s doping, respectively) typically requires
a) b)
ratherthickslabstoachieveconvergence(thisisspecially x2 x2
true for dielectrics with long screening lengths), (ii) an s]
nit
appropiateboundaryconditionatthe bottomoftheslab u
b.
which pins the chemical potential, µ, within the gap re- ar
gardlessoftheselectedpolytypeisaprerequisitetomake z [ x1/5
any differences in the doping among them meaningful +0.45 ++00..5655 x1/5
and,(iii)thepolarcharacterofSiC(111)/(0001)oriented
slabsintroducesanadditionalelectricfieldacrossthe di- δVH [eV] Energy [eV] δVH [eV] Energy [eV]
electric which may considerably alter the final doping
level. Additionally, the large dispersion of the π bands FIG. 1. (a) left: δVH(z) profile along a G/H/(SiC)6/H slab
depicted at the center and, right: the corresponding DOS
forming the Dirac cones requires a hyperfine sampling
projected, in ascending order, on thebottom H layer,thesix
of the Brillouin Zone (BZ) for an accurate estimation of
SiC bilayers, the intercalated H layer and the G; the bonds
the doping charge (see below), thus increasing consider- between C-terminated bottom of the slab and H atoms have
ably the computational time. Paradoxically, the associ- been elongated by 0.45 ˚A with respect to their optimized
ated low G-projected density of states (PDOS) around values (1.56 vs 1.11 ˚A). Black and blue lines correspond to
the DP makes the DP shift with respect to µ, ∆DP, a 3C-SiC(111) and 2H-SiC(0001), respectively. (b) Same as in
highly precisegaugefor the Gdoping,speciallyfromthe (a), but a G/H/(SiC)6/Au2 slab with gold instead of H ter-
mination is shown; the C-Au and Au-Au bonds have been
experimental side.
elongated by 0.55˚A and 0.65˚A respectively. The depicted
The first goal of this work is therefore to set up a cal- slab geometries in each figurecorrespond tothe3C-SiC(111)
culation strategy that overcomes the above fundamental case.
drawbacks. After considering different slab models we
have found that a G/H/(SiC)n/Au2 slab with a double SP induced doping– Let us first address the problems
goldlayerattached to the lowerC-danglingbonds meets outlinedaboverelatedtotheuseofaslabgeometrywhen
satisfactorilythe aboverequirements. The model is then trying to estimate the G doping. For the most com-
employedtocalculatetheQFGdopingsofdifferentpoly- mon practice of capping the lower C-terminated surface
typesasafunctionofthenumberofSiCbilayers(BLs)in with another hydrogen layer these shortcomings become
the slab, n. Apart from the cubic 3C-SiC(111), we have patent. In Figure 1(a) we plot the 2D averaged Hartree
considered those with largest hexagonality, namely, 2H- potential profiles, V (z), and the layer resolved PDOS
H
, 4H- and 6H-SiC(0001), having a stacking defect (SD) for a G/H/(SiC) /H slab assuming a 3C- (black lines)
6
every two, three and four BLs, respectively. The com- or2H-SiC(blue)stacking. Sinceconsiderabledipole mo-
bined effect of SP and defect-induced self-doping will be ments were found in both cases (∆V =0.8 and 1.5 eV
addressed at the end of this letter. for the 3C and 2H slabs, respectively) we elongated the
Methods– All presented calculations have been per- C-H bonds with respect to their optimized values (see
formedwiththeGREENcode[30]anditsinterfacetothe Fig. 1(a)) thus reducing the potential drops by as much
DFT basedSIESTApackage[31]. We employedthe gen- as 0.7 eV. Unfortunately, and regardless of this expan-
eralized gradient approximation [32] and included semi- sion, the DP remains pinned at the chemical potential
empirical van der Waals interactions [33] to account for for both slabs. The reason is the absence of any gap
the weak G/H/SiC interaction. Dipole-dipole interac- states at the bottom of the slab, so that charge neutral-
tions between spurious slab replicas were removed via ityatthegraphenelayerlocksµattheDP.Anaturalway
theusualdipole-dipolecorrections[34](DDC)thuslead- to overcome this drawback is to introduce states within
ing to vanishing electric fields in the vacuum. The rest the gap and localized at the bottom part of the slab. If
of the calculation parameters are described in detail in their associated DOS is much larger than that of the G,
Ref.[35]. TheG/H/(SiC) /Au modelslabscompriseda thentheyshouldpintheslab’schemicalpotentialjustas
n 2
(2 2)graphenelayerplacedontopofn(√3 √3)R30◦ bulk dopants/defects fix µ in a real surface, leaving the
× ×
SiC(111)/(0001) bilayers (BLs) with each Si dangling G’s Dirac cones free to trap or release electrons in order
bondattheuppersurfacesaturatedbyahydrogenatom, to screen any internal fields. Attaching a double gold
while the bottom C dangling bonds were saturated by layer at the C-ended lower layer does indeed yield the
3
desired boundary condition. In Fig. 1(b) we present the gible value of 14 meV for cubic 3C-SiC(111), the graphs
potential profiles and PDOS of such a G/H/(SiC) /Au clearly reveal how the DP shifts increase with the level
6 2
slab for 3C- and 2H-SiC. The potential drops (∆V=0.3 of hexagonality attaining several hundreds of meV.
and 0.5 eV) were reduced by 0.2 eV after expanding the Wenowturnourattentiononhowthesedopingsevolve
Au-C and Au-Au interlayer distances by large amounts withthefilmthickness. Figure2(b)illustratesthedepen-
(>0.5 ˚A). We stress that although the expanded bottom dence of the graphene doping charge δσ (left axis) and
geometry is not realistic, its purpose is to modify the lo- ∆DP (right axis) as a function of the slab thickness and
cal dipole at the bottom of the slab in order to reduce the polytype considered. For the 6H and 4H cases we
the total slab’s dipole moment. Inspection of the PDOS have defined different subsets of data depending on the
show that metal induced gap states (MIGSs) appear in location of the SD closest to the surface, which may be
both cases as a large broad peak mainly localized at the either between the first and second BLs (4H and 6H )
1 1
interface between the last SiC BL and the gold plane, or the second and third (4H and 6H ) or the third and
2 2
penetrating around three BLs into the substrate. The fourth(6H ). Exceptforthe3Ccase,whichremainswith
3
MIGSs now determine the slab’s chemicalpotential and, anegligibledopingforallthicknesses,therestofplotsre-
notably,thedopingintheQFGlayerhasalmostvanished veal an overall increase of δσ with n approaching their
forthe 3C case(blackline), while the 2Hslab(blue) dis- respective bulk SP values (indicated by thick horizontal
plays a clear p-doping with a considerable DP shift of lines at the right of the plot). The 6H and 4H polytypes
around250meV. We havealso consideredattaching sin- show a stair-like behavior which can be understood by
gleandtriplegoldlayersatthebottomoftheslabtofind noting that as n is increased by one, the added BL may
that the former (latter) yields a slight n (p)-type doping or may not introduce an additional SD in the slab. In
in the G [35]. Such slabs may therefore be employed as the former case,the slab’s SP increasesand δσ showsan
modelsystemswherebulkdopantsalreadyinducecertain abrupt raise, whereas in the latter case, the crystalline
level of doping [24]. regionincreasesandhence,thedepolarizationismoreef-
ficientleading to a slightdecreaseofthe doping. For the
(a) 0.5 (b) 2Hcase,ontheotherhand,δσshowsasmoothbehaviour
3C 6H1
V] with no jumps due to the absence of crystalline regions.
Energy [ey [eV]−000...055 4H121440 2H 312803 12σ [ ]2e x10 /cm 4H2 2H4H1 000...234∆DP [eV] HalatehntnoewcneeSae=ivrbCleyar1nV,2sdtaBh(tmsseuerwSaeaxPitFliilmofioganur.lsmot2ohf((caiδsVo)σ)npB,todMrsluiyob)etuytbtthpoeyaettttiohshfoeestrohccelrhlaaoesrrsmcggsreieinercetgahtnlhoainpiftcgokwtthnoeeeefnfisttvsniheaades-l
Energ 0.0 δ 3C6H1 6H2 6H3 −000.1.1 SPT.he QFG doping dependence displayed in Fig. 2 con-
−0.5 stitutesoneofthecentralresultsofthisworkasitclearly
Γ K M Γ K M n establishes a relationship between the SP and QFG p-
doping also providing an estimate for the slab size re-
FIG. 2. (a) DOS(k,E) projected on G, H and first three
quired to reach the bulk SP limit; at the largest thick-
SiC BLs of a G/H/SiC semi-infinite surface calculated after
ness considered, n = 12, the doping amounts to 60-80%
matching the Green’s function of a G/H/(SiC)12/Au2 slab
with that of the corresponding SiC bulk. Four different SiC of the SP while, from extrapolation, we expect that al-
polytype surfaces are shown: 3C, 6H1, 4H1 and 2H. ∆DP is most a 100% should be already reached at n & 20 BLs.
indicated in each plot in meV. (b) Doping of the graphene Simple electrostatic arguments dictate that the differ-
layerforall G/(SiC)n/Au2 slabsconsidered inthiswork asa ence between δσ and the SP can be entirely attributed
functionofn,theSiCpolitypeandthelocationoftheSDclos-
to the macroscopic electric field within the dielectric, E
est to the surface. Left axis gives the doping surface charge
density,δσ,andrightaxistheDiracpointshift,∆DP=DP−µ (generally known as the depolarization field) via [35]:
ǫ ǫ E =δσ+SP.
(non-linearscale). Thickhorizontallinesattherightindicate 0 r
thebulk SP associated to each polytype. Afurther remarkableissue inFig.2(b) is the factthat
an ample range of dopings can be obtained in an al-
The above analysis provides clear evidence that our most continuous way by controlling the number of SDs,
slab model allows to address the impact that the SDs their density and their proximity to the surface G layer.
present in the film have on the graphene layer. Fig- This is in line with the wide spread of experimental val-
ure 2(a) shows a precise picture of the Dirac cones in ues reported for the graphene’s doping within the same
the form of PDOS(k,E) for a 3C, 6H, 4H, and 2H sub- polytype. For instance, for the most commonly used
strate. They have been calculated for a semi-infinite 6H-SiC(0001) substrate, our calculated dopings fall in
geometry after replacing the Hamiltonian of the lower the 4 6 e 1012/cm2 range, in excellent agreement
− ×
half of the G/H/(SiC) /Au slab by that of the corre- with a number of experimental results varying from 2.0
12 2
sponding bulk polytype [38][39]. Apart from the negli- to 6.2 1012 cm−2 [6, 19, 21, 24]. Smaller δσ values
×
4
For the 3C cubic case, a considerable hole concentration
of 20.2 e 1012/cm2 (∆DP=208 meV) is induced in the
×
π bands in order to compensate the electron charge ac-
cumulatedinthevicinityofthe defect[12]. Asexpected,
thedopingisevenlargerforthe2Hslab,∆DP=300meV
or31.3e 1012/cm2, since the SP further contributesto
×
the doping. Most importantly, the difference in δσ be-
tweenbothpolytypesis11e 1012/cm2,whichcoincides
×
verynicelywiththeSP-induceddopingof10e 1012/cm2
×
found for undefected 2H slabs at n = 6 (blue line in
FIG. 3. Top (a) and side (b) view of the geometry corre- Fig. 2(b)) indicating that the SP- and self-doping mech-
sponding to a vacancy embedded in an (8×8) supercell of anisms are basically additive. This is not a trivial result
G/H/2H-SiC(0001)/Au2. In(a)onlygrapheneatomsandthe as it is not clear a priori if the self-doping mechanism
H atom saturating the dangling bond are shown. DOS(~k,E)
willaffectthedepolarizationfieldE which,asmentioned
projectedonthefirstlayersofthesurfacefor3C-SiC(111)(c)
above,playsanimportantroleinthefinalGdoping. Our
and 2H-SiC(0001) (d). Doping levels are indicated in meV.
results thus suggest that the self-doping, already very
largeonits own,hardlyaltersthe incomplete compensa-
tion of the SP pseudo-charge by the G’s π bands.
( 2 e 1012/cm2) have also been reported [8, 9, 14]
∼ × In summary, we have presented a methodology within
which, in light of our results, might be associated with
the framework of standard DFT slab calculations that
defects or impurities/dopants either at the G layer or
accounts for the contribution of the SiC substrate’s SP
in the bulk [24], or even to a cubic termination at the
to the QFG doping. The scheme relies on the pinning of
surface [14]. On the other hand, few values of δσ have
theslab’schemicalpotentialatthebottomoftheslabvia
beenreportedfora4H-SiC(0001)substrate: Mammadov
the creation of MIGSs states after saturating the lower
et al [24] obtained 6.9 and 8.6 e 1012/cm2 for an n-
× C-danglingbondswithagoldbilayer. Onemayaddition-
typedopedandasemi-insulatingsubstrate,respectively,
ally incorporate self-doping contributions due to defects
which is in very good accordance with our values for the
or adsorbates in the G layer. Application to QFG on
4H at n & 12, while the Hall measurements of Ur-
1/2 6H-,4H- and2H-SiC(0001)substratesindicates that full
ban et al [17] yielded charges of 15 20 e 1012/cm2
− × compensationoftheSPbythe Gdopingshouldoccurat
which signifcantly exceed the bulk SP, again suggesting
thicknesses of n & 20 BLs, while thinner slabs yield an
the presence of other sources of doping. Unfortunately,
ample range of dopings depending on the polytype and
wearenotawareofanyQFGexperimentscarriedouton
the precise termination of the surface. Other sources
a 2H surface.
of doping, such as self-doping in the G layer, or bulk
Competition between SP and self-doping– Finally, and dopants, may also be incorporated into the calculations.
in order to reacha generalpicture of the doping in QFG Apart from the obvious applicability of the analysis to
systems,weincorporatedefectsintheGlayerwithinour ultrathin SiC films, the scheme should also work satis-
goldterminated slabmodel. Among the various types of factorily in other dielectrics exhibiting an SP although
pointdefectsstudiedintheliterature[12,25,40]wecon- the nature of the metallic layer and the interlayer ex-
sider the Jahn-Teller distorted vacancy structure where pansions required to minimize the surface dipole will in
two C dangling bonds establish a bond among them in- general need to be tuned for the specific system.
ducing a pentagonal structure, while the third C dan-
J.S. acknowledges Polish Ministry of Science and
gling bond is saturated by a H atom (see Figure 3(a)).
Higher Education for financing the postdoctoral stay at
We makethis choicebecausethis defectstructurehardly
the ICMM-CSIC in the frame of the fellowship Mobility
alters the Dirac cones and, at the same time, yields a
Plus. H.A. and J.C. acknowledge financial support from
considerable p-type doping [12]. Due to computational
the Spanish Ministry of Innovation and Science under
limitations,the defectis embeddedinan(8 8)Gsuper-
contract Nos. MAT2012-38045-C04-04 and MAT2013-
×
cell corresponding to a somewhat large concentration of
47878-C2-Rrespectively.
0.8% (typical experimental values obtained for EG sam-
plesareoftheorderof0.006%),whilethe numberofSiC
BLs was set to n=6, leading to a total of 848 atoms in
the unit cell (see Fig. 3(b)).
PDOS(~k,E) maps are shown in Figs. 3(c) and (d), [1] J.-S. Moon, D. Curtis, M. Hu, D. Wong, C. McGuire,
P. Campbell, G. Jernigan, J. Tedesco, B. VanMil,
for the 3C and 2H polytypes, respectively. Although
R. Myers-Ward, C. Eddy, and D. Gaskill, Electron De-
a band gap of 150 meV appears between the Dirac vice Letters, IEEE 30, 650 (2009).
∼
bands in both cases due to the large defect concentra- [2] Y.-M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B.
tion, it is still possible to estimate their doping levels. Farmer, H.-Y. Chiu, A. Grill, and P. Avouris, Science
5
327, 662 (2010). [20] A. Drabinska, M. Kaminska, A. Wolos, W. Strupinski,
[3] Y. Wu, K. A. Jenkins, A. Valdes-Garcia, D. B. Farmer, A. Wysmolek, W. Bardyszewski, R. Bozek, and J. M.
Y.Zhu,A.A.Bol, C. Dimitrakopoulos, W.Zhu,F. Xia, Baranowski, Phys. Rev.B 88, 165413 (2013).
P.Avouris, andY.-M.Lin,NanoLetters12,3062(2012). [21] S. Rajput, M. Chen, Y. Liu, Y. Li, M. Weinert, and
[4] Y.-M. Lin, A. Valdes-Garcia, S.-J. Han, D. B. Farmer, L. Li, Nat Commun 4, 0 (2013).
I. Meric, Y. Sun, Y. Wu, C. Dimitrakopoulos, A. Grill, [22] J. Ristein, S. Mammadov, and T. Seyller, Phys. Rev.
P.Avouris, andK.A.Jenkins,Science332,1294(2011). Lett. 108, 246104 (2012).
[5] A. Tzalenchuk, S. Lara-Avila, A. Kalaboukhov, S. Pao- [23] S. Davydov,Semiconductors 46, 1186 (2012).
lillo, M. Syvajarvi, R. Yakimova, O. Kazakova, [24] S. Mammadov, J. Ristein, R. J. Koch, M. Ostler,
T. Janssen, V. Falko, and S. Kubatki, Nat. Nanotech- C.Raidel,M.Wanke,R.Vasiliauskas,R.Yakimova, and
nology 5, 186 (2010). T. Seyller, 2D Materials 1, 035003 (2014).
[6] F.Speck,J.Jobst, F. Fromm,M. Ostler,D.Waldmann, [25] O. V. Yazyev and L. Helm, Phys. Rev. B 75, 125408
M. Hundhausen, H. B. Weber, and T. Seyller, Applied (2007).
Physics Letters 99, 122106 (2011). [26] Q. Wang, W. Zhang, L. Wang, K. He, X. Ma, and
[7] S.Goler, C. Coletti, V.Piazza, P. Pingue, F. Colangelo, Q. Xue, Journal of Physics: Condensed Matter 25,
V.Pellegrini,K.Emtsev,S.Forti,U.Starke,F.Beltram, 095002 (2013).
and S. Heun,Carbon 51, 249 (2013). [27] P. Mallet, F. Varchon, C. Naud, L. Magaud, C. Berger,
[8] J. A. Robinson, M. Hollander, M. LaBella, K. A. Trum- and J.-Y. Veuillen,Phys. Rev.B 76, 041403 (2007).
bull, R. Cavalero, and D. W. Snyder, Nano Letters 11, [28] Y. Qi, S. H. Rhim, G. F. Sun, M. Weinert, and L. Li,
3875 (2011). Phys. Rev.Lett. 105, 085502 (2010).
[9] C. Riedl, C. Coletti, T. Iwasaki, A. A. Zakharov, and [29] L. Simon, C. Bena, F. Vonau, D. Aubel, H. Nasrallah,
U.Starke, Phys.Rev.Lett. 103, 246804 (2009). M.Habar, andJ.C.Peruchetti,TheEuropeanPhysical
[10] C. Coletti, K. V. Emtsev, A. A. Zakharov, T. Ouisse, Journal B 69, 351 (2009).
D. Chaussende, and U. Starke, Applied Physics Letters [30] J.Cerda´,M.A.VanHove,P.Sautet, andM.Salmeron,
99, 081904 (2011). Phys. Rev.B 56, 15885 (1997).
[11] S. Forti and U. Starke, Journal of Physics D: Applied [31] J. M. Soler, E. Artacho, J. D. Gale, A. Garcia, J. Jun-
Physics 47, 094013 (2014). quera, P. Ordejon, and D. Sanchez-Portal, Journal of
[12] J. Slawin´ska and J. I. Cerda´, Carbon 74, 146 (2014). Physics: Condensed Matter 14, 2745 (2002).
[13] S. Tanabe, Y. Sekine, H. Kageshima, and H. Hibino, [32] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.
JapaneseJournalofAppliedPhysics51,02BN02(2012). Lett. 77, 3865 (1996).
[14] S. Forti, K. V. Emtsev, C. Coletti, A. A. Zakharov, [33] F. Ortmann, F. Bechstedt, and W. G. Schmidt, Phys.
C.Riedl, andU.Starke,Phys.Rev.B84,125449(2011). Rev. B 73, 205101 (2006).
[15] S. Watcharinyanon, C. Virojanadara, J. Osiecki, A. Za- [34] J. NeugebauerandM. Scheffler,Phys.Rev.B 46,16067
kharov, R. Yakimova, R. Uhrberg, and L. Johansson, (1992).
Surface Science 605, 1662 (2011). [35] J. Slawin´ska, H. Aramberri, M. Mun˜oz, and J. Cerda´,
[16] S. Rajput, Y. Y. Li, and L. Li, Applied Physics Letters to be published,arXiv:1410.4551 (2014).
104, 041908 (2014). [36] A. Qteish, V. Heine, and R. Needs, Physica B: Con-
[17] J.M.Urban,P.Dabrowski,J.Binder,M.Kopciuszynski, densed Matter 185, 366 (1993).
A.Wysmolek,Z.Klusek,M.Jalochowski,W.Strupinski, [37] A.Qteish,V.Heine, andR.J.Needs,Phys.Rev.B45,
and J. M. Baranowski, Journal of Applied Physics 115, 6376 (1992).
233504 (2014). [38] E. T. R.Rossen, C. F. J. Flipse, and J. I. Cerda´, Phys.
[18] C. Coletti, S. Forti, A. Principi, K. V. Emtsev, A. A. Rev. B 87, 235412 (2013).
Zakharov, K. M. Daniels, B. K. Daas, M. V. S. Chan- [39] Thesemi-infiniteapproachdoesnotmodifytheDPshifts
drashekhar, T. Ouisse, D. Chaussende, A. H. MacDon- deduced from the slab calculations since the bulk em-
ald, M. Polini, and U. Starke, Phys. Rev.B 88, 155439 ployed in the matching process does not contribute to
(2013). the SP.See also Ref.[35].
[19] J. C. Johannsen, S. Ulstrup, M. Bianchi, R. Hatch, [40] F. Banhart, J. Kotakoski, and A. V. Krasheninnikov,
D.Guan,F.Mazzola,L.Hornekar,F.Fromm,C.Raidel, ACS Nano5, 26 (2011).
T. Seyller, and P. Hofmann, Journal of Physics: Con-
densed Matter 25, 094001 (2013).