the effect of atomic-scale defects and dopants on graphene...
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IntroductionHydrogen adsorptionBandgap engineering
Summary
The effect of atomic-scale defects and dopantson Graphene electronic structure
Rocco Martinazzo
Dip. di Chimica-Fisica e ElettrochimicaUniversita’ degli Studi di Milano, Milan, Italy
GraphITAGSNL, Assergi (L’Aquila), May 15th - 18th 2011
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Outline
1 Introduction
2 Hydrogen adsorption
3 Bandgap engineering
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Outline
1 Introduction
2 Hydrogen adsorption
3 Bandgap engineering
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Outline
1 Introduction
2 Hydrogen adsorption
3 Bandgap engineering
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Outline
1 Introduction
2 Hydrogen adsorption
3 Bandgap engineering
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Ionic binding
-3 -2 -1 0 1 2 3( ε−εF ) / eV
0
15
30Cl
-3 -2 -1 0 1 2 3( ε−εF ) / eV
0
15
30KDonor level
Acceptor level
δεF δεF
DOSs are unchaged except for donor/acceptor levelselectron / hole dopingAtomic species are mobileLi, Na, K, Cs.. vs Cl,Br,I,..
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Covalent binding
-3 -2 -1 0 1 2 3( ε−εF ) / eV
0
15
30F
-3 -2 -1 0 1 2 3( ε−εF ) / eV
0
15
30V2
Midgap states show up in the DOSsAtomic species are immobileH, F, OH, CH3, etc. behave similarly to vacancies
See e.g., T. O. Wehling, M. I. Katsnelson and A. I. Lichtenstein, Phys. Rev. B 80, 085428 (2008)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Vacancies vs adatoms
-3 -2 -1 0 1 2 3( ε−εF ) / eV
0
15
30V3
-3 -2 -1 0 1 2 3( ε−εF ) / eV
0
15
30V2
-3 -2 -1 0 1 2 3( ε−εF ) / eV
0
15
30VH
Unrelaxed Relaxed Saturated
See e.g., F. Banhart, J. Kotakoski, A. V. Krasheninnikov, ACS Nano 5, 26 (2011)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Vacancies vs adatoms
High-energy e−/ion beams⇒ Random arrangement
Low-energy beams (kinetic control)⇒ Clustering due to preferential sticking
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Outline
1 Introduction
2 Hydrogen adsorption
3 Bandgap engineering
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Hydrogen chemisorption on graphene
Sticking is thermally activated1,2
Midgap states are generated upon stickingDiffusion does not occur3,4
Preferential sticking and clustering3,5,6
[1] L. Jeloaica and V. Sidis, Chem. Phys. Lett. 300, 157 (1999)[2] X. Sha and B. Jackson, Surf. Sci. 496, 318 (2002)[3] L. Hornekaer et al., Phys. Rev. Lett. 97, 186102 (2006)[4] J. C. Meyer et al., Nature 454, 319 (2008)[5] A. Andree et al., Chem. Phys. Lett. 425, 99 (2006)[6] L. Hornekaer et al., Chem. Phys. Lett. 446, 237 (2007)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Sticking
1 2 3 4z / Å
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
E / e
V
L. Jeloaica and V. Sidis, Chem. Phys. Lett. 300, 157 (1999)X. Sha and B. Jackson, Surf. Sci. 496, 318 (2002)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Midgap states
..patterned spin-density
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Midgap states
HTB =�
σ,ij(tija†i,σbj,σ + tjib
†j,σai,σ)
Electron-hole symmetry
bi → −bi =⇒ h → −h
if �i is eigenvalue andc†i
=P
iαi a
†i
+P
jβj b
†j
eigenvector
⇓
−�i is also eigenvalue andc
�†i
=P
iαi a
†i−
Pjβj b
†j
is eigenvector
9>>>=
>>>;n∗
nA + nB − 2n∗9>>>=
>>>;n∗
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Midgap states
HTB =�
τ,ij(tija†i,τbj,τ + tjib
†j,τai,τ )
TheoremIf nA > nB there exist (at least) nI = nA − nB "midgap states" with vanishing
components on B sites
Proof.»
0 T†
T 0
– »αβ
–=
»00
–with T nBxnA( > nB)
=⇒ Tα = 0 has nA − nB solutions
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Midgap states
HHb =�
τ,ij(tija†i,τbj,τ + tjib
†j,τai,τ ) + U
�ini,τni,−τ
TheoremIf U > 0, the ground-state at half-filling has
S = |nA − nB |/2 = nI/2
Proof.E.H. Lieb, Phys. Rev. Lett. 62, 1201 (1989)
...basically, we can apply Hund’s rule to previous result
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Midgap states for isolated “defects”
M.M. Ugeda, I. Brihuega, F. Guinea and J.M. Gomez-Rodriguez, Phys. Rev. Lett. 104, 096804 (2010)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Midgap states for isolated “defects”
ψ(x , y , z) ∼ 1/r
V. M. Pereira et al., Phys. Rev. Lett. 96, 036801 (2006);Phys. Rev. B 77, 115109 (2008)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Dimers
S. Casolo, O.M. Lovvik, R. Martinazzo and G.F. Tantardini, J. Chem. Phys. 130 054704 (2009)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Dimers
[1] [2]
[1] L. Hornekaer, Z. Sljivancanin, W. Xu, R. Otero, E. Rauls, I. Stensgaard, E. Laegsgaard, B. Hammer and F.Besenbacher. Phys. Rev. Lett. 96 156104 (2006)
[2] A. Andree, M. Le Lay, T. Zecho and J. Kupper, Chem. Phys. Lett. 425 99 (2006)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Clusters
µ = 1µB ⇒µ = 2µB⇒ µ = 3µB
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Clusters
A2B A3
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Role of edges
zig-zag edge sites haveenhanced hydrogen affinitygeometric effects can beinvestigated in smallgraphenes
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Role of edges
imbalanced ‘PAHs’ balanced PAHs
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Role of edges
imbalanced ‘PAHs’ balanced PAHs
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Role of edges: graphenic vs edge sites
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Role of edges: graphenic vs edge sites
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Outline
1 Introduction
2 Hydrogen adsorption
3 Bandgap engineering
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Band-gap opening
Electron confinement: nanoribbons, (nanotubes),etc.Symmetry breaking: epitaxial growth, deposition, etc.Symmetry preserving: “supergraphenes”
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Band-gap opening
Electron confinement: nanoribbons, (nanotubes),etc.Symmetry breaking: epitaxial growth, deposition, etc.Symmetry preserving: “supergraphenes”
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Band-gap opening
Electron confinement: nanoribbons, (nanotubes),etc.Symmetry breaking: epitaxial growth, deposition, etc.Symmetry preserving: “supergraphenes”
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
e-h symmetry
HTB =�
σ,ij(tija†i,σbj,σ + tjib
†j,σai,σ)
Electron-hole symmetry
bi → −bi =⇒ h → −h
if �i is eigenvalue andc†i
=P
iαi a
†i
+P
jβj b
†j
eigenvector
⇓
−�i is also eigenvalue andc
�†i
=P
iαi a
†i−
Pjβj b
†j
is eigenvector
9>>>=
>>>;n∗
nA + nB − 2n∗9>>>=
>>>;n∗
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Spatial symmetry
r-space
σd
σv
AC CB
C6 2I C
G0 = D6h
⇒ G(K) = D3h
k-space
σd
C3
K
K K
G(k) = {g ∈ G0|gk = k + G}⇒ G(K) = D3h
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Spatial symmetry
r-space
σd
σv
AC CB
C6 2I C
G0 = D6h
⇒ G(K) = D3h
k-space
C2v
D3h
D2h
sC
D6h
G(k) = {g ∈ G0|gk = k + G}⇒ G(K) = D3h
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Spatial symmetry
Γ K Μ Γ
-10.0
-5.0
0.0
5.0
10.0
(ε−ε
F) / e
V
NN onlyTB fit to DFT data
D6h D3h
C2vD2h
A2u B2
E"
B2g
A2B2g
C2v
B2
B2
B1uA2
B2
C2v
B2g
A2u
|Ak� = 1√NBK
PR∈BK
e−ikR |AR�
|Bk� = 1√NBK
PR∈BK
e−ikR |BR�
�r |AR� = φpZ(r− R)
For k = K (or K�)
{|Ak� , |Bk�} span the E �� irrep of D3h
Degeneracy is lifted at first order (noi symmetry in D3h)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Spatial and e-h symmetry
A E
n even even even or odd
Lemmae-h symmetry holds within each kind of symmetry
species (A, E, ..)
TheoremFor any bipartite lattice at half-filling, if the number
of E irreps is odd at a special point, there is a
degeneracy at the Fermi level, i.e. Egap = 0
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
A simple recipe
Consider nxn graphene superlattices (i.e. G = D6h):degeneracy is expected at Γ, KIntroduce pZ vacancies while preserving point symmetryCheck whether it is possible to turn the number of E irrepsto be even both at Γ and at K
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Counting the number of E irreps
n = 4
K: EK: 2A + 2E : 2A: 2A + 2E ΓΓ
Γ A E0̄3 2m
2 2m2
1̄3 2(3m2 + 2m + 1) 2(3m
2 + 2m)
2̄3 2(3m2 + 4m + 2) 2(3m
2 + 4m + 1)
Kn A E0̄3 2m
2 2m2
1̄3 2m(3m + 2) 2m(3m + 2) + 12̄3 2(3m
2 + 4m + 1) 2(3m2 + 4m + 1) + 1
⇒ n = 3m + 1, 3m + 2, m ∈ N
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
An example
(14, 0)-honeycomb
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Band-gap opening..
Tight-binding
�gap(K ) ∼ 2t√
1.683/n
DFT
�gap(K ) ∼ 2t√
1.683/n
R. Martinazzo, S. Casolo and G.F. Tantardini, Phys. Rev. B, 81 245420 (2010)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
..and Dirac cones
..not only: as degeneracy may still occur at � �= �F
new Dirac points are expected
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
..and Dirac cones
..not only: as degeneracy may still occur at � �= �F
new Dirac points are expected
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Antidot superlattices
...the same holds for honeycomb antidots
1 nm
3 nm0 10 20 30 40 50
n0.0
0.1
0.2
0.3
0.4
0.5
ε gap /
eV
d = 1 nmd = 2 nmd = 4 nm
Γ K M Γ0.0
0.2
0.4
ε gap /
eV
0 2 4 6 8 10 12 14an / nm
0.0
0.1
0.2
0.3
0.4
0.5
ε gap /
eV
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Antidot superlattices
...the same holds for honeycomb antidots
M. D. Fishbein and M. Drndic, Appl. Phys. Lett. 93,113107 (2008)
T. Shen et al. Appl. Phys. Lett. 93, 122102 (2008)
J. Bai et al. Nature Nantotech. 5, 190 (2010) J. Bai et al.
Nature Nantotech. 5, 190 (2010) J. Bai et al. Nature
Nantotech. 5, 190 (2010)
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Summary
Covalently bound species generate midgap species uponbond formationMidgap states affect chemical reactivityThermodynamically and kinetically favoured configurationsminimize sublattice imbalanceSymmetry breaking is not necessary to open a gap
Atomic-scale defects on graphene
IntroductionHydrogen adsorptionBandgap engineering
Summary
Acknowledgements
University of Milan
Gian Franco Tantardini
Simone Casolo
Matteo Bonfanti
Chemical Dynamics Theory Group
http://users.unimi.it/cdtg
+-x:
C.I.L.E.A. SupercomputingCenterNoturI.S.T.M.
Atomic-scale defects on graphene