This study uses density functional theory calculations with a hybrid functional approach to compare graphene epitaxial layers on cubic and hexagonal silicon carbide polytypes. The results show that while the Dirac point shifts relative to the bulk bands with different polytypes, the essential Dirac feature of the graphene layer is preserved. A uniform Fermi velocity and similar carrier doping levels are found. Additionally, a small substrate-mediated band gap of 25-40 meV is induced at the Dirac point for all polytypes due to the broken symmetry of the graphene-SiC interface.

1. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Graphene on Cubic and Hexagonal SiC:
A Comparative Theoretical Study
O. Pankratov S. Hensel P. Götzfried M. Bockstedte
Lehrstuhl für Theoretische Festkörperphysik
DPG Frühjahrstagung
Regensburg
2013

2. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Graphene on Cubic and Hexagonal SiC
Motivation
→ SiC exhibits variety of different polytypes
c-axis 3C
A B C A
0
zinc blende
hexagonality
2H
A B
1
wurzite
4H
A B C
1/2
cubic
hexagonal
6H
A B C A
1/3

3. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Graphene on Cubic and Hexagonal SiC
Questions and Results (Si-face)
1 Influence of substrate polytypes on Epilayer properties?
2 Dirac band alignment relative to bulk bands?
Preservation of essential epilayer Dirac feature
Substrate-mediated Dirac band gap (εg = 25 ∼ 40 meV)
ED, EF shift systematically relative to VBM

4. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Graphene on Cubic and Hexagonal SiC
Calculational Method
Problem 1) Band gap deficiency of LDA-functional
Combined LDA-/HSE hybrid functional approach
(Heyd-Scuseria-Ernzerhof)
1
◦ Admix fraction of nonlocal HF-type exchange
EHSE
xc = EPBE
xc + 1
4
EHF
x − EPBE
x
◦ Screened interaction reduces calculational expense
1
r
≡ erfc(ωr)
r
short-range
+ erf(ωr)
r
long-range
Density functional program package VASP2
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
[1] Heyd et al., J.Chem.Phys. (2003) [2] Kresse and Furthmüller, PRB (1996)

5. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Graphene on Cubic and Hexagonal SiC
Structural Access
Problem 2) Incommensurable lattices
Si-face (6
√
3×6
√
3)R30 buffer layer1
C-face no buffer,2,3 mutually rotated layer4
(
√
3×
√
3)R30 Model
computationally tractable
correct description of graphene on Si-face5,6
◦ covalent interface bonding
◦ correct epilayer distance
◦ strained graphene interface has minor effect7
⇒ Focus on the Si-face
PRL 99, 076802 (2007)
[1] Otha et al., Science (2006) [2] Emtsev et al., PRB (2008) [3] Hiebel et al., PRB (2009)
[4] Haas et al., PRL (2008) [5] Mattausch et al., PRL (2007) [6] Varchon et al., PRL (2007)
[7] Pankratov et al., PRB (2010)

6. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Graphene Epilayer on SiC (Si-face)
Origin of Band Structure
Dirac feature emerges clearly
Charge density distribution ρ(z) = |Ψ(x, y, z)|2
dxdy
Evac = common energy reference (LDA/HSE)
ED, EF and interface state position model-independent
(similar for LDA and HSE)

7. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Graphene Epilayer on SiC (Si-face)
Origin of Band Structure
Dirac feature emerges clearly
Charge density distribution ρ(z) = |Ψ(x, y, z)|2
dxdy
Evac = common energy reference (LDA/HSE)
ED, EF and interface state position model-independent
(similar for LDA and HSE)

8. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Graphene Epilayer on SiC (Si-face)
Dirac Point Position
ED, EF shift systematically with hexagonality
HSE in good agreement with experiments
|EF − ED| ∼= const. ⇒ ne− 5.6 · 1013
cm−2
SiC EHSE
D − EHSE
V [eV] EExp
D − EExp
V [eV] EF − ED [eV]
3C 1.48 0.62
6H 1.90 2.25 [1] 0.63
4H 2.53 2.91±±±0.1 [2] 0.61
2H 2.39 0.62
[1] Ristein et al., PRL 108, 246104 (2012) [2] Sonde et al., PRB 80, 241406 (2009)

9. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Graphene Epilayer on SiC (Si-face)
Dirac Band Gap
Dirac band gap εg = 25 ∼ 40 meV same for HSE and LDA
Similar small εg for strain-free 5x5 structure1
Upper experimental limit given by ARPES resolution (40 ∼ 50 meV)
SiC HSE
g [meV]
LDA
g [meV]
3C 41.4 36.7
6H 39.9 35.4
4H 28.0 26.3
2H 33.9 32.9
[1] Pankratov et al., PRB 82, 121416R (2010)

10. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Interlude: Origin of Dirac Band Gap
→ Isolated graphene: C3v = {E, 2c3, 3σ}
C3 ⇒ HK =
0∗ V∗ V∗
V∗
0∗ V∗
V∗ V∗
0∗
, V = |V| eiϕ
∈ C
ε1 = 2|V| cos ϕ ε2,3 = 2|V| cos 2π
3 ± ϕ
3σ-mirror planes ⇒ ϕ fixed ⇒ ggg = 000

11. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Interlude: Origin of Dirac Band Gap
→ Isolated graphene: C3v = {E, 2c3, 3σ}
C3 ⇒ HK =
0∗ V∗ V∗
V∗
0∗ V∗
V∗ V∗
0∗
, V = |V| eiϕ
∈ C
ε1 = 2|V| cos ϕ ε2,3 = 2|V| cos 2π
3 ± ϕ
3σ-mirror planes ⇒ ϕ fixed ⇒ ggg = 000
→ AB-stacked Epilayer: HK + v for |v| |V|
No mirror planes but τττ-displaced planes
Composite matrix elements acquire phase
shift |V| + |v|e
2π
3
i

12. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Interlude: Origin of Dirac Band Gap
→ Isolated graphene: C3v = {E, 2c3, 3σ}
C3 ⇒ HK =
0∗ V∗ V∗
V∗
0∗ V∗
V∗ V∗
0∗
, V = |V| eiϕ
∈ C
ε1 = 2|V| cos ϕ ε2,3 = 2|V| cos 2π
3 ± ϕ
3σ-mirror planes ⇒ ϕ fixed ⇒ ggg = 000
→ AB-stacked Epilayer: HK + v for |v| |V|
No mirror planes but τττ-displaced planes
Composite matrix elements acquire phase
shift |V| + |v|e
2π
3
i
AB-stack: No mirror planes ⇒⇒⇒ Dirac gap
g 3|v|, coupling strength |v| 10 meV
Relativistic massive particle spectrum
AA-stack: ⇒ g = 0⇒ g = 0⇒ g = 0 (rel. massless spectrum)

13. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Interlude: Origin of Dirac Band Gap
→ Isolated graphene: C3v = {E, 2c3, 3σ}
C3 ⇒ HK =
0∗ V∗ V∗
V∗
0∗ V∗
V∗ V∗
0∗
, V = |V| eiϕ
∈ C
ε1 = 2|V| cos ϕ ε2,3 = 2|V| cos 2π
3 ± ϕ
3σ-mirror planes ⇒ ϕ fixed ⇒ ggg = 000
→ AB-stacked Epilayer: HK + v for |v| |V|
No mirror planes but τττ-displaced planes
Composite matrix elements acquire phase
shift |V| + |v|e
2π
3
i
AB-stack: No mirror planes ⇒⇒⇒ Dirac gap
g 3|v|, coupling strength |v| 10 meV
Relativistic massive particle spectrum
AA-stack: ⇒ g = 0⇒ g = 0⇒ g = 0 (rel. massless spectrum)

14. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Graphene Epilayer on SiC (Si-face)
Fermi Velocity
Relativistic massive Epilayer spectrum near ED
E(k; g , vF) =
g
2
2
+ ( k vF)2 g →0
−−−−→ k vF
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)
⇒ near ED: Dirac states almost free of substrate interaction (besides small g)

15. INTRODUCTION METHOD & MODEL RESULTS SUMMARY
Summary - Graphene on Different SiC Polytypes
Ab initio Calculations with Combined LDA and HSE06 Hybrid Functional Approach
1 Influence of substrate polytypes on Epilayer?
2 Dirac band alignment relative to bulk bands?
Preservation of essential epilayer Dirac feature
Uniform Fermi-velocity, similar n-doping (Fermi-pinning)
Substrate-mediated Dirac band gap1
(εg = 25 ∼ 40 meV)
ED, EF shift systematically relative to VBM
HSE06 - good agreement with experimental data, LDA - correctly
corroborates trends
[1] Pankratov et al., PRB 86, 155432 (2012)

18. Appendix: Buffer Layer on SiC (Si-face)
No Dirac bands near EF
Half-filled interface state pins Fermi level
Interface state shifts upwards with CB
HSE - asymmetric band gap opening

19. Appendix: Epilayer Doping (Si-face)
n-doped epilayer
carrier density approximation
ne−/e+ =
|EF − ED|2
π( vF )2
universal Fermi-pinning mechanism yields
polytype-independent ne−
∼= 5.6 · 1013
cm−2