The effect of atomic-scale defects and dopants on Graphene ...graphita.bo.imm.cnr.it/graphita2011/GraphITA_PDF_Talks/Martinazzo... · The effect of atomic-scale defects and dopants

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


Summary

Outline

1 Introduction

2 Hydrogen adsorption

3 Bandgap engineering



Summary

Outline

1 Introduction





Summary

Outline

1 Introduction





Summary

Outline

1 Introduction





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,..



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)



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)



Summary

Vacancies vs adatoms

High-energy e−/ion beams⇒ Random arrangement

Low-energy beams (kinetic control)⇒ Clustering due to preferential sticking



Summary

Outline

1 Introduction





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)



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)



Summary

Midgap states

..patterned spin-density



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∗



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



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



Summary

Midgap states for isolated “defects”

M.M. Ugeda, I. Brihuega, F. Guinea and J.M. Gomez-Rodriguez, Phys. Rev. Lett. 104, 096804 (2010)



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)



Summary

Dimers



Summary

Dimers

S. Casolo, O.M. Lovvik, R. Martinazzo and G.F. Tantardini, J. Chem. Phys. 130 054704 (2009)



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)



Summary

Clusters

µ = 1µB ⇒µ = 2µB⇒ µ = 3µB



Summary

Clusters

A2B A3



Summary

Role of edges

zig-zag edge sites haveenhanced hydrogen affinitygeometric effects can beinvestigated in smallgraphenes



Summary

Role of edges

imbalanced ‘PAHs’ balanced PAHs



Summary

Role of edges

imbalanced ‘PAHs’ balanced PAHs



Summary

Role of edges: graphenic vs edge sites



Summary

Role of edges: graphenic vs edge sites



Summary

Outline

1 Introduction





Summary

Band-gap opening

Electron confinement: nanoribbons, (nanotubes),etc.Symmetry breaking: epitaxial growth, deposition, etc.Symmetry preserving: “supergraphenes”



Summary

Band-gap opening




Summary

Band-gap opening




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∗



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



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



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)



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



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



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



Summary

An example

(14, 0)-honeycomb



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)



Summary

..and Dirac cones

..not only: as degeneracy may still occur at � �= �F

new Dirac points are expected



Summary

..and Dirac cones

..not only: as degeneracy may still occur at � �= �F

new Dirac points are expected



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



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)



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



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.



Summary

Acknowledgements

Thank you for your attention!


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The effect of atomic-scale defects and dopants on Graphene ...graphita.bo.imm.cnr.it/graphita2011/GraphITA_PDF_Talks/Martinazzo... · The effect of atomic-scale defects and dopants