The document investigates the impact of silicon carbide (SiC) polytype on the electronic structure of graphene epitaxial layers. It finds that the position of the Dirac point and Fermi level systematically shift with increasing hexagonality of the SiC polytype. Additionally, a gap of 30-40 meV is observed at the Dirac point, independent of the polytype. Low energy phonons at the SiC/graphene interface are also found to cause electron scattering by varying the separation between graphene and buffer layers, reducing graphene layer conductivity compared to suspended graphene. Calculations using the Boltzmann formalism are able to reproduce the experimentally observed temperature dependence of conductivity for different Fermi energies.

1. Abstract
We have investigated the impact of the SiC polytype on the epilayer electronic structure (left hand side panel below). Due to the similar surface structure of all polytypes the graphene-SiC interface
should be largely polytype-independent. However, the bulk electronic bands of the polytypes differ substantially as reflected by a notable band gap variation from 3.33 eV for 2H-SiC to 2.39 eV for
3C-SiC. Hence the alignment of the Dirac bands with the bulk SiC energy spectrum should vary substantially for different polytypes. We find that the Dirac point systematically shifts with respect to
the valence band edge with increasing polytype hexagonality; as the pinned Fermi level follows the same trend hence the intrinsic n-type doping remains almost the same. We further find a model
independent weak binding on the C-face and a gap at the Dirac point of 30-40 meV independent of the polytype.
Regarding the electronic transport properties, the presence of the SiC substrate (including the buffer layer) is known to cause a decrease of the graphene epilayer conductivity. We describe a
mechanism in which low energy phonons of the SiC/graphene interface modify the separation between the buffer layer and the graphene overlayer, resulting in a deformation potential and hence
charge carrier scattering (right hand side panel below). We determine this deformation potential ab-initio, and then calculate transport properties within the Boltzmann formalism. Our results can
reproduce well the experimentally observed temperature-dependence of the graphene epilayer conductivity.
Interface electronic structure of graphene on SiC
Motivation and open questions
SiC polytypes What is the influence of different substrate polytypes and interface
models on the graphene epilayer properties?
What is the exact Dirac band alignment relative to the bulk bands?
Confrontation with two problems: the band gap deficiency of the LDA-
functional and lattice incommensurability of graphene and SiC.
Structural models and calculational method
The simplified (
√
3×
√
3)R30 model correctly describes
graphene on the Si-face.1
It accommodates a strained
2×2 graphene cell and exhibits a single dangling bond.
The 5x5 buffer layer is a medium-scale, naturally oc-
curring2
and strain-free model. It accommodates a ro-
tated as well as bernal stacked graphene epilayer.
Both interface models share a covalently bonded buffer layer and emerging Dirac cone with the second C-layer.
combined LDA-/HSE3
hy-
brid functional approach
Density functional program package VASP4
XC-functional L(S)DA HSE06 hybrid
Cut-off energy [eV] 520 eV 420 eV
Γ-centered k-sampling 9x9x1 6x6x1
PAW pseudo-potentials CA PBE
Impact of SiC polytype on graphene epilayer
The position of ED, EF and the interface related
state systematically shift relative to VBM with
increasing hexagonality. (However, the positions
relative to Evac are similar for LDA and HSE data points.)
SiC EHSE
D − EHSE
V [eV] EExp
D − EExp
V [eV] EF − ED [eV] HSE
g [meV] LDA
g [meV]
3C 1.48 0.62 41.4 36.7
6H 1.90 2.25[1]
0.63 39.9 35.4
4H 2.53 2.91±±±0.1[2]
0.61 28.0 26.3
2H 2.39 0.62 33.9 32.9
|EF − ED| ∼= const. ⇒ constant epilayer doping ne− 5.6 · 1013
cm−2
Dirac band gap εg = 25 ∼ 40 meV same for HSE and LDA
vF = 0.84 v0
F uniform Fermi velocity
v0
F = 8.33 · 105 m/s (isolated strain-free graphene),
v∗
F = 0.87 v0
F (isolated 8 % strained graphene)
Analytical spectrum near ED (PRB 86, 155432 (2012))
Beside the small gap εg the Dirac states are almost free of substrate interaction.
HK + v =
0∗ w∗ w∗
w∗
0∗ w∗
w∗ w∗
0∗
, w = |V| + |v|e
2πi/3
|V|eiγ
εg 3|v| ≈ 30 meV
E(k; εg, vF) =
εg
2
2
+ ( k vF)2 εg→0
−−−→ k vF
The absence of mirror planes for the AB-stacked epilayer leads to the Dirac band splitting
(spectrum of relativistic massive particles).
5×5 buffer layer model
asymmetric in-
terface bonding
weaker residual substrate
interaction at the C-face
Analytical spectrum near ED
(PRB 82, 121416(R) (2010))
HDirac =



−∆ p∗
2b 0
p ∆ 0 0
2b∗
0 −∆ −p∗
0 0 −p ∆



εi(k)
(i=1,...,4)
= ±|b| (±) ∆ |b|
2
+ k vF
2
32.2◦
twisted epilayer bernal type
|b| ∆ [meV] |b| ∆
2.1 2.1 C-face 20.3 29.9
12.7 21.7 Si-face 13.5 11.0
On the Si-face the Dirac gap is about 30 meV. At certain energies, the Dirac bands are distorted
due to resonant interaction with interface states, which should lead to mobility suppression.
[1] Mattausch and Pankratov, PRL 99, 076802 (2007); Varchon et al., PRL 99, 126805 (2007)
[2] Riedl et al., PRB 76, 245406 (2007)
[3] Heyd et al., J.Chem.Phys. (2003); Paier et al., J.Chem.Phys. 124, 154709 (2006)
[4] Kresse and Furthm¨uller, PRB 54, 11169 (1996)
Graphene layer conductivity1
Motivation
0
20000
40000
60000
80000
100000
120000
Mobility[cm
2
/(Vs)]
suspended graphene
graphene on SiO2
graphene on SiC(0001)
The extremely high conductivity measured for (freestanding)
graphene is drastically reduced2
in the case of epitaxial
graphene on SiC. We can explain this effect with the following
phonon induced scattering mechanism.
Mechanism
s
graphene
buffer
SiC
Transverse graphene-buffer interface
phonons cause a change in the distance
s between graphene layer and buffer
layer.
According to DFT band structure calcula-
tions3
a variation of s causes a variation
of the relative position ∆ = EF −ED of the
Dirac cone.
This perturbation of the Dirac bands re-
sults in electron scattering between the
Dirac electron eigenstates. Hence the
graphene layer conductivity is reduced.
Calculation
The probability wk,k+q of elastic electron scattering from state |k to state |k + q can be
calculated from Fermi’s Golden Rule
wk,k+q =
2π
δ( k+q − k)
Φf
k + q, Φf
d∆
ds
δs k, Φi
2
.
k is the energy of the electron state |k , and Φi (Φf )
denotes the phonon distribution before (after) scatter-
ing. In this expression we insert the value d∆/ds =
150 meV/ ˚A extracted from the DFT band structure
calculations (5 × 5 model). The displacement δs
of the graphene layer relative to the buffer layer is
due to transverse phonons and can be expressed via
creation/annihilation-operators as
δs =
2A
q
1
√
ωq
eiq rˆaq + e−iq rˆa†
q ,
where A is the mass of the graphene layer. Note, that
the results will strongly depend on the dispersion ωq
of the participating phonon mode. For the described
mechanism the most important mode is the “breathing
mode” between buffer and graphene layer.
To calculate the conductivity σ from the scattering
probability wk,k+q we apply the Boltzmann transport
formalism and the relaxation time approximation (c.f.
eqs., vF is the Fermi velocity and f0
k is the Fermi distri-
bution function).
kx
ky
k
q=k'-k
k'
θk'
0 0.05 0.1
q (Å
-1
)
0
0.1
0.2
0.3
0.4
h
_
ωq
(meV)
0 0.05 0.1
k (Å
-1
)
0
0.5
1
τk
(10
-13
s)
ωq
= ω0
+ κ q
2
ωq
= (ω0
2
+ (C q)
2
)
1/2
ωq
= ω0
ωq
= C q
0 200 400 600
Ek
(meV)
~ ω0
2
k
-1
~ C
2
k
1
τk
=
q
(1 − cos θk+q) wk,k+q
σ =
e2
v2
F
2π
k dk −
∂f0
k
∂ k
τk
Results
40 80 120
EF
(meV)
0
1
2
3
4
ρ(kΩ)
0 100 200 300 400
EF
(meV)
0
5
10
15
20
25
ρ(kΩ)
T = 50 K
T = 100 K
T = 200 K
T = 300 K
T = 2 K
For this calculation we used
ωq = ω2
0 + C2q2, where ω0 = 0.1 meV and C = 500 m/s.
τ = τ−1
k + τ−1
0
−1
, where τ0 = 0.6 · 10−13
s accounts for
temperature-independent scattering.
In agreement with experiment4
we find differ-
ent temperature-dependencies of the resis-
tivity ρ = σ−1
for two regimes:
For Fermi energies far from the Dirac
energy, ρ depends linearly on T. This
reflects the increasing phonon popula-
tion with increasing T, and hence the in-
creased scattering.
For Fermi energies near the Dirac en-
ergy, the T-dependence of ρ is inverse.
This is driven by the increase of the
number of charge carriers (electrons and
holes) with increasing T.
[1] N. Ray et al., Phys. Rev. B 86, 125426 (2012)
[2] Nature Nanotech. 3, 206 (2008); PRL 101, 096802 (2008); APL 95, 122102 (2009); PRB 81, 195434 (2010); PRB 84, 115458 (2011)
[3] Pankratov et al., Phys. Rev. B 82, 121416(R) (2010)
[4] Jobst et al., PRB 81, 195434 (2010); Tanabe et al., PRB 84, 115458 (2011)
Lehrstuhl f¨ur Theoretische Festk¨orperphysik - FAU Erlangen - http://www.tfkp.physik.uni-erlangen.de oleg.pankratov@physik.uni-erlangen.de