Graphene on Insulating Oxide Substrates: Role of Surface Dangling States

0.38 MB·English

by  Priyamvada Jadaun

Graphene on Insulating Oxide Substrates: Role of Surface Dangling States
∗
Priyamvada Jadaun, Sanjay K. Banerjee, and Bhagawan Sahu
Microelectronics Research Center, The University of Texas at Austin, Austin Texas 78758
(Dated: January 28, 2010)
We study the effect of insulating oxide substrates on the energy band structure of monolayer
and bilayer graphene using a first principles density functional based electronic structure method
and a local exchange correlation approximation. We consider two crystalline substrates, SiO2 (or
α-quartz) and Al2O3 (α-alumina or sapphire), each with two surface terminations. We focus on
0 the role of substrate surface dangling states and their passivation in perturbing the linear energy
1 spectrum of graphene. On non-passivated surface terminations, with the relaxation of top surface
0 layers, only Si-terminated quartz retains the linear band structure of graphene due to relatively
2 large equilibrium separation from the graphene layer whereas the other three surface terminations
considerably distort it. However, without relaxations of the top surface layer atoms, linear bands
n
appear in the electronic spectrum but with the Dirac point shifted away from the Fermi level.
a
Interestingly, with a second carbon layer on non-passivated oxygen terminated Quartz, with top
J
surface layers relaxation, graphenefeatures appear in thespectrum but sapphire with both surface
8
terminations shows perturbed features even with two carbon layers. By passivating the surface
2
dangling states with hydrogen atoms and without top layer atomic relaxations, the electron-hole

symmetry occurs exactly at the Fermi level. This suggests that surface dangling states play a less
]
i importantrolethantheatomicrelaxationsofthetopsurfacelayersindistortingthelinearspectrum.
c
Inall cases we findthat thefirst layerofgraphene forms ripples, muchlikein suspendedgraphene,
s
- but the strength of rippling is found to be weaker probably due to the presence of the substrate.
l We discuss the energetics of both passivated and non-passivated surface terminations with a top
r
t graphenelayer.
m
. PACSnumbers: 71.15.Mb,71.20-b,73.20.-r
t
a
m
I. INTRODUCTION structures that result from the interaction of graphene
-
d with these surface terminations.
n
The excitement in graphene research, due to the real-
o
c ization of table-top high-energy physics experiments and Experimentally,therearestudieswhichattempttoun-
[ the promise to replace silicon in future semiconductor derstand the atomic structure of graphene on insulat-
chips, is certainly overwhelming1. Advancement in un- ingsubstrates10,usingnanometerscalemicroscopictech-
2
niques such as scanning tunneling and atomic force mi-
derstandingthefundamentalphysicsofelectronandhole
v
9 transport in it2 and using graphene for spintronics3 is croscopies. In addition, using Raman spectroscopy, the
role of substrates on phonon dispersions is also reported
9 critical to the realization of carbon-based electronics.
9 However, to make a transition from graphene science which indirectly hints at the change in the electronic
2 to engineering and finally to a viable technology, it is spectrumofgrapheneduetothepresenceofsubstrates11.
0. crucial to understand the interaction of graphene with Moroever, we are aware of two recent density functional
1 external parameters such as substrates4, contacts for studies of monolayer graphene on a crystalline SiO2
9 measurements5, and the role of high dielectric constant substrate12,13. Due to different and insufficient struc-
0 oxides6inadditiontostudyingtheeffectsofintermediate tural details in those reports, we could not compare our
v: species which are present in a particular process flow of resultswiththeconclusionsreachedinReferences12and
i graphene device fabrication. These parameters can pose 13.
X
a fundamental challenge to the ultimate realization of
r
a carbon electronics. In this article, we report the role of Ourpaperisorganizedasfollows. InsectionII,wefirst
one of these externalparameters,namely the substrates, discuss the computational method and the convergence
in changing the electronic structure of graphene using a parametersusedforthisstudyfollowedbythemotivation
density functional based electronic structure method7,8 and the procedure for building the surface models from
and a local approximation to the exchange-correlation theircorrespondingbulkcounterparts. Ourresultsofthe
potential(LDA)9. Weusetwoinsulatingcrystallinesub- effects of dangling state and its passivationon the linear
strates, SiO (or quartz) and Al O (or sapphire), each spectrum of graphene and the role of atomic relaxation
2 2 3
with two surface terminations in our study. These are is discussed in section III. We describe the band struc-
widelyusedsubstratesinexperimentsthatuseexfoliated turewiththehelpoforbitalandatomprojecteddensities
graphene. Weconsidersurfacedanglingstatepassivation of states. We then present the energetics of graphene on
with hydrogen atoms and atomic relaxations and focus bothnon-passivatedandpassivatedsurfaceterminations.
on their role in perturbing the graphene electronic spec- Finally, we summarize our results and present our con-
trum. We discuss the energetics and the energy band clusions.
2
4
(a)
0
V)
e
y(
g
r
ne (b)
E
2
0
−2
FIG.1: (Coloronline)Schematicillustrationofthesupercell
−4
structure of monolayer graphene on the top of (a) Si and Γ M K Γ
(b) Oxygen-terminated quartz. Four unit cells of quartz are
shownwithSiatomsinblue,oxygenatomsinredandcarbon
FIG.2: EnergybandstructuresofmonolayergrapheneonSi-
atoms in yellow. Each Si atom is surrounded by four oxygen
terminated quartz, at high-symmetry points in the supercell
atoms.Thesupercellstructureofgrapheneonthetopofboth Brillouin zone, at the interlayer distances of (a) 3 ˚A and (b)
Alandoxygenterminated sapphireissimilar exceptthatthe 2.5 ˚A. The atoms in the supercell were not relaxed and the
successive Al and oxygen layers are stacked vertically at an
top surface was not passivated. The interlayer distance cor-
angle. Hence we donot show them here.
reponds to the location of graphene plane from the topmost
Si-plane in the Si-terminated quartz substrate. The Fermi
energy is set at zero. The origin of occurance of linear bands
attheΓ-pointinsteadofK-pointofthesupercellisexplained
II. COMPUTATIONAL METHOD AND
in the text.
SURFACE MODELS
This section addresses the details of the computa- We find that the optimized lattice constants of quartz
tional method we used followed by the procedure we and sapphire are close to the experimental values (Ta-
adopted to obtain the surface models of quartz and ble I). The in-plane and out-of-plane lattice constants
sapphire from their bulk counterparts and the conver- for quartz and sapphire differ, from the experimental
gence parameters used for this study. We used a plane- values, by less than 1 %. Using these optimized lat-
wave based electronic structure method with local den- tice parameters, we constructed surface models of both
sity approximation(LDA)7 for exchange and correlation quartzandsapphireas follows. Fourbulk unitcells were
and the projector augmented plane-wave potential for stacked along the c-direction (which corresponds to a
electron-ion interaction8. The bulk structures of both thin-film thickness of about 22 ˚A) and we find that 6
the quartz and sapphire are consistent with those avail- × dC−C graphene, containing a total of 24 carbon (C)
ableinliterature14 andthesurfacemodelsbuiltfromthe atoms (where dC−C =1.42 ˚A) is nearly commensurate
bulk structuresconformto the widely acceptedα-quartz with the hexagonal surface of the substrates (Fig. 1).
and α-alumina structures15. We first optimized the lat- The lattice mismatch of the quartz and sapphire termi-
tice parameters of bulk crystalline quartz and sapphire. nations with graphene is calculated to be 0.19 % and
For both the bulk substrates, we generated hexagonal 0.42 % respectively. We note that the commensurability
unit cells from the originalrhombohedralcells and these of graphene with the underlying substrate is not neces-
bulk phases form layered structures (alternating cation sary for our study since it is not focussed on epitaxial
and anion layers). The unit cell of quartz contains 27 growth of graphene on SiC and other metal substrates.
atoms with 3 Si planes and 6 oxygen planes, each plane Therefore, the lattice mismatch values mentioned here
containing3atoms,whereastheunitcellofsapphirecon- serve only as an initial guideline in assessing the degree
tains30atomswith4aluminum(Al)planesand6oxygen of distortion of strictly two-dimensional graphene in the
planes again each plane containing 3 atoms. We used a presence of the substrates. In fact, we observe rippling
7 × 7 × 5 k-point mesh in the hexagonal Brillouin zone of graphenelayerson eachof the surface terminations in
(BZ) and a 612 eV kinetic energy cut-off. The results the final relaxed structures. We note that the atomic-
were carefully checked with respect to a larger k-point scaleroughnesssimulatedherebyrelaxingtopfewlayers
set and higher energy cut-offs. of substrates does not resemble the roughness present
3
on amorphous SiO2 (or a-SiO2) and Al2O3 (or a-Al2O3) III. ENERGY BAND DISPERSIONS,
surfacesthatareusedinexperimentsinvolvinggraphene. DENSITIES OF STATES AND ENERGETICS
Simulating roughnessof amorphous surfaces,using DFT
basedelectronicstructuremethod,willincreasethecom-
In this section, we discuss the role of dangling states,
putationalburdensignificantlyduetotherequirementof their passivation and the role of atomic relaxations on
large supercell size which is necessary to achieve rough-
the electronic band structure of graphene on the sur-
ness at the macroscopic scale. However, in the surface faceterminationsweconsideredhere. Wenotethatboth
models adopted here, in their final equilibrium configu-
the surface terminations of quartz possess large number
rations, some roughness is present at the atomic scale. of dangling states whereas in sapphire, by construction
Periodic boundary conditions were enforced along the of the surface models from bulk, the number of nearest
surface directions whereas a vacuum size of 10 ˚A was neighbors of both Al and oxygenis found to be optimal.
used along the c direction to enable periodic slab calcu- Therefore, for passivation studies, only quartz surfaces
lations. The silicon(Si) danglingstates atthe bottomof were considered. We passivated the dangling states with
the supercell were saturated with hydrogen atoms. We hydrogen atoms. There are four different possibilities
fixedthesupercelllatticeparametersinallcasesandonly that emerge from considering whether surface dangling
the atoms in the planes of the top two unit-cells of the states are passivated or not and whether or not atomic
substrate and the atoms in the graphene planes were al- relaxations are performed.
lowed to relax. We used the same energy cut-off as in
WefirstdiscussSi-terminatedquartz. Wedidnotpas-
the bulk calculations but the k-point mesh in the BZ
sivate the dangling states and did not relax the atomic
was chosen to be 7 × 7 × 1. For atomic relaxations,the
positions. To get a rough estimate of the interlayer sep-
totalenergywasassumedtohaveconvergedwhenallthe
aration at which graphene electronic structure is mini-
componentsoftheHellman-Feynmanforcesweresmaller
mally perturbed, we placed the graphene layer at vari-
than 0.01 eV/˚A.
ous distances from the top substrate layer. The choice
of these distances and distances chosen for other surface
terminations, are arbitrary. We chose bilayer graphene
interlayer distance (which is 3.34 ˚A) as a reference. At
(a) a distance of 3 ˚A and above, from the top Si surface,
4 graphene retains the linear spectrum but at a smaller
distance (of 2.5 ˚A), the linear bands are perturbed (Fig.
3
2(a)and (b)). The occurance of the linear spectrum at
2 the Γ-point ofthe supercellBZ instead ofthe K-pointis
1 due to crystalsymmetry of the supercell and its relation
to the monolayergraphene lattice symmetry. The origin
0 of such shifting of the location of the electron-hole sym-
metryfromtheK-pointtotheΓ-pointisseenincalcula-
V) tionsinvolvingsub-monolayeralkalimetaladsorptionon
e
gy( (b) graphene surfaces21. We discuss the origin of this shift
r
ne in the Appendix section. On relaxing the atomic posi-
E
tions of the top few layers of the quartz substrate and
graphene, we get an equlibrium distance of 3 ˚A between
2
the top substrate layer and graphene due to which the
1 perturbationto electronicspectrumis minimal,asinthe
non-relaxed case (Figure not shown).
0
We now discuss Al-terminated sapphire and its inter-
action with graphene. For the non-passivated surface
−1
without atomic relaxations, the dispersion curve shows
−2 the danglingstates atthe Fermilevelareresonancewith
Γ M K Γ
the carbon pz orbital at interlayer distances as high as
4 ˚A and as low as 2.7 ˚A (Fig. 3(a) and (b)). However,
FIG. 3: Energy band structures of monolayer graphene on
withatomicrelaxations,wegetanequlibriumseparation
Al-terminated sapphire, along high-symmetry points in su-
percellBrillouinzone,atinterlayerdistancesof(a)4.0˚Aand of 2.7 ˚A and a similar energy dispersion curves (Figures
(b) 2.7 ˚A. The atoms in the supercell were not relaxed and now shown).
the top surface was not passivated. The interlayer distance The situation is similar in both oxygen-terminated
correponds to the location of the graphene plane from the quartz and sapphire. On non-passivated oxygen-
topmost Al-plane in Al-terminated sapphire substrate. The
terminated quartz without atomic relaxations, at dis-
Fermi energy is set at zero. tances of 1.76 ˚A and 2.5 ˚A, graphene features appear in
the band structure (Fig. 4(a) and (b)) but the electron-
hole symmetry is located above the Fermi level but an
4
4
(a) (a)
4
2 0
0
−3
V)
e
y( −2 (b)
Energ 4 (b) y(eV) 2
g
r
e
n
E
0
2
−2
(c)
0 1
0
−2
Γ M K Γ −1
FIG. 4: Energy band structures of monolayer graphene on −2
oxygen-terminatedquartz,atainterlayerdistanceof(a)1.76
˚Aand(b)2.5˚A.Theatomsinthesupercellwerenotrelaxed
and the top surface was not passivated. The interlayer dis- Γ M K Γ
tance correponds to the location of the graphene plane from
the topmost oxygen plane in oxygen-terminated quartz sub- FIG.5: Energybandstructuresofmonolayergrapheneon(a)
strate. The Fermi energy is set at zero. oxygen-terminated quartz, (b) oxygen-terminated sapphire,
alonghigh-symmetrypointsin supercellBrillouin zone,com-
putedat therespective equilibrium separations shown in Ta-
energy gap opens up for the case of 2.5 ˚A. Opening of ble I and (c) two carbon layers on the top of the oxygen-
terminatedquartzseemstorestorethelinearbehavioratthe
an energy gap may be due to the breaking of sub-lattice
Fermi level. The atoms in thesupercell were relaxed but the
symmetryingrapheneasobservedinepitaxialgrowthof
top surface was not passivated. The Fermi energy is set at
grapheneonSiCsubstrate22 butismostlikelyduetoap-
zero.
proximations in DFT. On relaxing the atomic positions,
oxygen-terminatedsurfacesinboth quartzandsapphire,
maintain an equilibrirm separation which is quite close equilibrium distances are larger than that in the corre-
to the graphene layer. As a result, we find a strong per- sponding interatomic distances in the bulk or molecular
turbation to the linear spectrum of graphene (Fig. 5(a) phases,itsuggeststhatgraphenedoesnotformabulk-or
and (b)). molecular-like phase with the surface terminations used
When we add a second carbon layer on oxygen- in this study.
terminatedquartz andrelax the atomic positions,we re- We now discuss the effect of passivation of dangling
coverlinearspectrumofgraphene(Fig. 5(c)). Thishints statesonthebandstructureofgraphene. Sincegraphene
at the existence of a buffer layer in studies of graphene on non-passivated Si-terminated quartz already shows
on oxygen-terminated quartz. However, our calculations thealinearspectrum,weconsideredpassivatingonlythe
suggest more than two carbon layers are needed to re- danglingstatesofoxygen-terminatedquartz. Wedidnot
covergraphenelinearenergyspectrumincaseofAl- and relax the atomic positions. Linear bands seem to ap-
oxygen-terminated sapphire. pear above the Fermi level for distances below 2 ˚A but
These equlibrium distances, for each non-passivated right at the Fermi level in case of distances above 2 ˚A
surface terminations with atomic relaxations, are listed (Fig. 6(a) and (b)). It is likely that relaxing the atomic
inTable I.Italsolists interatomicdistancesbetweenthe positions may retain the graphene features but with a
Carbon and the corresponding atom of the surface ter- different equilibrium separation than the non-passivated
minations in the bulk or molecular phase. Since all the surface. We did not consider this case explicitly in our
5
calculations.
TABLEI: Latticeparameters(in˚A)ofbulkquartzandsap-
Our results suggest that presence or absence of dan- phire, average interplanar distances (in ˚A) of graphene from
gling states is not as effective in distorting the linear
the four underlying surface terminations and their binding
band structure as the atomic relaxations. It is the equi- energies (in eV/atom), in case of non-passivated but relaxed
librium distance that dictates the survival of graphene surfaces. Thenumbersinparenthesisaretheout-of-planelat-
features in the dispersion spectrum. However, a case in tice constants of the bulk phases and those in square paren-
which relaxations of passivated substrate and graphene thesisareinteratomicdistancesbetweenCarbonandthecor-
provide a favorable equilbrium distance for retention of responding atom of the surface terminations in the bulk or
the graphene features, is not ruled out. molecular phase. It should be noted that, for isolated sub-
stratecalculations,wepassivatedthetoplayerwithhydrogen
To understand the perturbations to the linear spec-
atoms to keep thesupercell non-magnetic.
trum of graphene in case of non-passivated relaxed sur-
face, we plot atom and orbital projected densities of Lattice Paramters d(C-x) Ebind
states (or DOS) (Fig. 7). The top panel (a) shows DOS (x= Si,Al,O)
Bulk quartz
for oxygen terminated quartz, projected on to s- and p-
This work 4.914 (5.408)
statesofoxygenatomofthetopsurfacelayerandcarbon. Expt.a 4.913 (5.405)
We seeastronghybridizationbetweenoxygen-pandC-p b
Si-terminated 3.0 (1.89) 10.061
orbitals in the vicinity of the Fermi level. A similar con- c
Oxygen-terminated 1.76(1.3) 0.581
clusion was reached in a recent DFT-based calculation
on monolayer graphene on oxygen-terminated quartz13.
Bulk sapphire
Such resonance structures are also seen in the DOS of This work 4.907 (4.908)
bothAl-andoxygen-terminatedsapphire(panels(b)and Expt.d 4.943 (4.907)
(c)). Thisexplainswhygrapheneπ-orbitals,afteratomic Al-terminated 2.7 (1.89-2.19)e 1.017
relaxations, cannot retain their linear dispersion in the Oxygen-terminated 2.15 0.689
presence of non-passivated surfaces. We also estimated
aReference14,16
average height fluctuations of two-dimensional graphene
bReference17
due to the presence of substrate and the atomic relax- cReference18
ations ofthe topsurface layers. Inallcases,we findthat dReference15,19
the deviations are ∼ 0.05 ˚A which is small compared to eReference20
suspended graphene23.
Table I shows the binding energy values for graphene
onthefoursurfaceterminationsconsideredinthisstudy.
In the most relevant case i.e. no surface passivation but IV. SUMMARY AND CONCLUSION
withatomicrelaxations,thesevalueshintatnon-bonding
nature of graphene to the underlying oxide substrates.
In summary, we have studied, using a first princi-
The binding energy values are obtained by using the fol-
ples DFT method, the effect of two crystalline insulat-
lowing definition.
ing substrates, quartz and sapphire, on the electronic
structure of monolayer graphene. We considered the ef-
fect of surface passivation and atomic relaxations on the
Ebind =E(supercell)−E(Gr)−E(substrate) (1) linear spectrum of graphene. Dangling states seems to
be ineffective in distorting the linear band structure of
grapheneexceptshiftingthespectrumaboveFermilevel.
where E(supercell) denotes the total energy of the su-
It is the atomic relxations which dictate the equlibrium
percell containing the substrate and a graphene layer.
separation between graphene and the topmost surface
E(Gr) and E(substrate) denote, respectively, the total
layer and this distance decides the strength of perturba-
energies of isolated graphene and isolated substrate in tion on linear bands. Si-terminated α-quartz retains the
the same supercell set-up, with the same energy cut-off
graphene band structure on non-passivated surface even
and k-point mesh as that of the combined graphene and
with atomic relaxations. This leaves oxygen-terminated
substrate calculations.
quartz surface where atomic relaxations play an impor-
However,we find that passivatedbut non-relaxedsur- tantrole. BothAl- andoxygen-terminatedsapphireper-
faces are energetically more favorable compared to both turbthelinearbandswhetherornottheirdanglingstates
non-passivated non-relaxed as well as non-passivated are passivated and whether or not atomic relaxations
relaxed surfaces. This suggests an existence of a are performed. Two carbon layers are necessary to re-
metastable configuration of graphene in presence of the coverthe linearbandstructureofgrapheneonthe topof
non-passivated surfaces (both with and without atomic oxygen-terminated quartz whereas our calculations hint
relaxations). Of course, passivated and relaxed surfaces, atmore than two carbonlayersare needed in caseof Al-
which we did not explicitly calculate here, can be lower and oxyen-terminatedSapphire. Energetically,graphene
in energy than any of these cases. on non-passivating surface terminations is metastable.
6
O-s
(a) O-p
C-s
4 C-p
2
(a)
E
F
2
1
0
V)
nergy(e −2 00.6 (b) AACCll----ppss
E
(b)
4 m)
E
o F
at
− 0.4
V
e
s/
2 ate
st
S( 0.2
O
D
0
0
O-s
(c) O-p
C-s
C-p
−2
Γ M K Γ
2 E
F
FIG. 6: (Color online) Same as Figure 4 but now for passi-
vated surface of oxygen terminated quartz.
1
Acknowledgments
The authors acknowledge financial support from the
0
DARPA-CERA and NRI-SWAN program. The authors −20 −16 −12 −8 −4 0 4
acknowledge the allocation of computing time on NSF Energy(eV)
TeragridmachinesRanger(TG-DMR080016N)andLon-
estar at Texas Advanced Computing Center. FIG. 7: (Color online) Atom and orbital projected density
of states for a C atom in monolayer graphene and the atoms
of the topmost planes in (a) oxygen-terminated quartz (b)
Al-terminated sapphire and (c) oxygen-terminated sapphire
Appendix
substratescomputedattheequilibrium separationsshown in
Table I. The atoms in the supercell were relaxed but thetop
Quartz and sapphire substrates and the graphene surface was not passivated. The Fermi energy is set at zero.
→ →
layer, all have a hexagonal unit cell. Let A , A and
→ 1 2
A be the lattice vectorsof the entire graphene and sub-
3 → → → tureofthegrapheneplussubstratesystembelow. Taking
strate system and B1, B2, and B3 be th→e c→orrep→onding →a1 to be along xˆ,
reciprocallatticevectors. Similarily,let(a , a , a )and
1 2 3
→ → →
(b , b and b ) be the triad of primitive and reciprocal
1 2 3 →
a =axˆ (A.1)
lattice vectors of the graphene layer alone. 1
Wenotethatthelatticestructureofthegrapheneplus
substrate system is a (2 p(3) x 2 p(3))R30o recon- → 1 p(3)
a =a( xˆ+ yˆ) (A.2)
struction of the lattice structure of the graphene layer 2
2 2
alone. Thus A = 2p(3)a where A is the lattice con-
stant of the entire system and a is the lattice constant
→ p(3) 1
of the two-dimensional graphene layer. Since both unit A =2p(3)a( xˆ+ yˆ) (A.3)
1
cells are hexagonal this implies that the reciprocal lat- 2 2
tice structure of the graphene layer on top is a (2p(3) x
→
2p(3))R30oreconstructionofthereciprocallatticestruc- A2 =2p(3)ayˆ (A.4)
7
→
A =cˆz (A.5)
3
→ 2π 1
b = (xˆ− yˆ) (A.6)
y 1 a p(3)
b
2
→ 2π 2
b = ( yˆ) (A.7)
2
a p(3)
K
→ 2π 2
B x B1 = ( xˆ) (A.8)
B 1 2p(3)a p(3)
2
Γ M b
1
→ 2π 1
B = (− xˆ+yˆ) (A.9)
2
2p(3)a p(3)
Figure 8 shows the BZ set-up with these vectors using
equations A.6 to A.9.
Now if we take the same Γ point as the center of both
FIG.8: (Coloronline)SchematicsofsupercellBrillouinzone
the Brillouin zones and draw the reciprocal unit cell of
(small)superposedonthegraphenehoneycombBrillouinzone
both the structures we realize that the K point of only
(large). The reciprocal lattice vectors are shown along with
the overlap of Γ to K vector of graphene with the B1 vector t→he graphene layer lies on the reciprocal lattice vector
of thesupercell. Theoverlap is duetothesymmetry reasons B of the graphene plus substrate system (as shown in
1
and is explained in thetext. figure). This means that the K point of graphene layer
folds in by symmetry onto the Γ point of the entire sys-
The reciprocal lattice vectors are, tem.
∗ Electronic address: [email protected] 8 G. Kresse and D.Joubert, Phys.Rev.B 59, 1758 (1999).
1 A. K.Geim, Science 324, 1530 (2009). 9 D. M. Ceperley and B. J. Alder, Phys. Rev.Lett. 45, 566
2 A. H. Castro Neto, F. Guinea, N. M. Peres, K. S. (1980);J.P.PerdewandA.Zunger,Phys.Rev.B23,5048
Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (1981).
(2009). 10 A. Deshpande, W. Bao, F. Miao, C. N. Lau and B. J.
3 C. Jozsa, M. Popinciuc, N. Tombros, H.T. Jonkman, and LeRoy, Phys. Rev. B 79, 205411(R) (2009); V. Geringer,
B. J. van Wees, Phys.Rev. B 79, 081402(R) (2009). M. Liebmann, T. Echtermeyer, S. Runte, M. Schmidt, R.
4 X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, Ruckamp,M.C.Lemme,andM.Morgenstern,Phys.Rev.
A. Valamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Lett 102, 076102 (2009); K. R. Knox et. al. Phys. Rev.
Colombo and R. S. Ruoff, Science 324, 1312 (2009); A. B 78, 201408 (2008); M. Ishigami, J. H. Chen, W. G.
Reina, X.Jia, J.Ho, D.Nezich,H.Son, B.Bulovic, M.S. Cullen, M. S. Fuhrer and E. D. Williams, Nano. Lett 7,
Dresselhaus and J. Kong, Nano Lett. 9, 30 (2009); K. S. 1643(2007); T.TsukamotoandT.Ogino,App.Phys.Ex-
Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, press 2, 075502 (2009); U. Stoberl, U. Wurstbauer, W.
J-H.Ahn,P.Kim,J-Y.Choi,andB.H.Hong,Nature457, Wegscheider, D. Weiss, and J. Eroms, Appl. Phys. Lett
706 (2009). 93, 051906 (2008).
5 J. Cayssol, B. Huard, and D. Goldhaber-Gordon, Phys. 11 I.Calizo,W.Bao,F.Miao,C.N.Lau,andA.A.Balandin,
Rev.B.79,075428(2009);S.Russo,M.F.Craciun,M.Ya- Appl. Phys. Lett. 91, 201904 (2007); A. C. Ferrari et. al.
mamoto,A.F.MorpurgoandS.Tarucha,arXiv:0901.0485; Phys. Rev.Lett 97, 187401 (2006).
R.Murali,Y.Yang,K.Brenner,T.BeckandJ.D.Meindl, 12 Y-J. Kang, J. Kang, and K. J. Chang, Phys. Rev. B 78,
Appl. Phys. Lett. 94, 243114 (2009); A. Venugopali, L. 115404 (2008);
Colombo and E. Vogel, Appl. Phys. Lett. 93, 013512 13 P. Shemella and Saroj K. Nayak, Appl. Phys. Lett 94,
(2010). 032101 (2009).
6 S. Kim, J. Nah, I. Jo, D. Shahrjerdi, L. Colombo, Z. Yao, 14 C. R. Helms and E. H. Poindexter, Rep. Prog. Phys. 57,
E.TutucandS.K.Banerjee,Appl.Phys.Lett.94,062107 791(1994);N.J.deLeeuw,F.M.HigginsandS.C.Parker,
(2009); A. Pirkle, R. M. Wallace and L. Colombo, Appl. J. Phys. Chem. B 103, 1270 (1999); G.-M.Rignanese,
Phys. Lett. 95, 133106 (2009). AlessandroDeVita,J.-C.Charlier,X.GonzeandRoberto
7 G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 Car, Phys.Rev. B 61, 13250 (2000).
(1996); G. Kresse and J. Furthmuller, Comput. Mat. Sci. 15 T.M.FrenchandG.A.Somorjai,J.Phys.Chem.74,2489
6, 15 (1996); J. Hafner, J. Comp. Chem. 29, 2044 (2008). (1970);C.BarthandM.Reichling,Nature414,54(2001).
8
16 http://cst-www.nrl.navy.mil/lattice/struk/sio2a.html. 22 S. Y, Zhou, G.-H, Gweon, A. V. Fedorov, P. N. First, W.
17 C. M. Zetterling, ”Process technology for Silicon Carbide A.derHeer,D.H.Lee.F.Guinea,A.H.CastroNetoand
Devices”,EMISprocedingSeries,No.2,INSPEC,IEEUK A. Lanzara, Nature Mater. 6, 770 (2007); S. Y. Zhou, D.
2002. A. Siegel, A. V. Fedorov, F. El. Gabaly, A. K. Schmid,
18 F. H. Allen, O. Kennard, D. G. Watson et. al., J. Chem. A. H. Castro Neto and A. Lanzara, Nature Mater. 7, 259
Soc. Perkin Trans. II, S1-S9(1987). (2008).
19 H. Bialas and H.J. Stolz, Z. Physik B 21, 319 (1975). 23 A.Fasolino,J.H.LosandM.I.Katsnelson,NatureMater.
20 G. A. Jeffrey and V. Y.Wu,Acta Cryst. 20, 538 (1966). 6, 858 (2007).
21 M. Farjam and H. Rafii-Tabar, Phys. Rev. B 79, 045417
(2009).