defective graphene, not so bad after all

Defective graphene, not so bad after all..

Rocco Martinazzo

Department of Physical Chemistry and ElectrochemistryUniversita’ degli Studi di Milano, Milan, Italy

Universität PotsdamDec. 17th 2010

(dcfe@university of milan) Defects in graphene Potsdam2010 1 / 44

Outline

1 Introduction

2 H atom adsorption energeticsClusters of H atoms

3 Bandgap engineeringSuperlattices of H atoms and the like


Introduction

Outline

1 Introduction




Introduction

H2 in ISM

Hydrogen is the most abundantelement of the UniverseH2 is formed on the surface of dustgrain

Hydrogen-graphite (graphene) is an important model forunderstanding H2 formation in ISM

fgrain=ngrain/nH ∼ 10−12 i.e. ∼1% ofISM mass


Introduction

Technology

Hydrogen storageNuclear fusionNanoelectronics, spintronics,nanomagnetism


H atom adsorption energetics

Outline

1 Introduction




H atom adsorption energetics

The need for understanding adsorption

H on Graphite (Graphene) vs metal substrates

Chemisorption is thermally activated1,2

Substantial lattice reconstruction upon sticking1,2

Diffusion of chemisorbed H atoms does not occur3

Preferential sticking3

Clustering of H atoms3,4,5

Dimer recombination6

[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] A. Andree et al., Chem. Phys. Lett. 425, 99 (2006) [5] L. Hornekaer et

al., Chem. Phys. Lett. 446, 237 (2007) [6] L. Hornekaer et al., Phys. Rev. Lett. 96, 156104 (2006)


H atom adsorption energetics Clustering

Single-H adsorption

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 /

eV

L. Jeloaica and V. Sidis, Chem. Phys. Lett. 300, 157 (1999)X. Sha and B. Jackson, Surf. Sci. 496, 318 (2002)



Substrate electronic structure

..patterned spin-density




..patterned spin-density




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



Properties of bipartite lattices

HTB =∑

σ,ij(tija†i,σbj,σ + tjib

†j,σai,σ)

Electron-hole symmetry

bi → −bi =⇒ H → −H

if εi is eigenvalue andc†i =

Pi αi a

†i +

Pj βj b

†j eigenvector

⇓

−εi is also eigenvalue andc

′†i =

Pi αi a

†i −

Pj βj b

†j is eigenvector

9>>>=>>>; n∗

nA + nB − 2n∗9>>>=>>>; n∗




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 Bsites

Proof. »0 T†T 0

– »αβ

–=

»00

–with T nBxnA( > nB)

=⇒ Tα = 0 has nA − nB solutions




HHb =∑

τ,ij(tija†i,τbj,τ + tjib

†j,τai,τ ) + U

∑i ni,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 (1989) 1201

...basically, we can apply Hund’s rule to previous result



Midgap states for isolated “defects”

ψ(x , y , z) ∼ 1/r



Dimers



Dimers

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



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)



3-atom clusters, etc.

A2B


Bandgap engineering

Outline

1 Introduction




Bandgap engineering Superlattices of H atoms and the like

Basic electronic structure

H ≈ −t∑

i,τ∑

j a†τ (Ri)bτ (Ri + δj) + c.c.

aτ ;i = 1√N

Pk e−ikRi aτ (k)

H = −tP

k,τ f (k)a†τ (k)bτ (k) + c.c.

H = −tXk,τ

ha†τ (k), b†τ (k)

i »0 f (k)

f∗(k) 0

– »aτ (k)bτ (k)

–



Technology



Device related properties

Thickness: thinnest gate-controlled regions in transistorsMobility: high-mobility carriersHigh-field transport: high saturation velocitiesBand-gap: high on-off ratios are not achievable without abandgap



Logic applications

nS=ε0εVgt e

CNT-FET with ordinary and wrapped aroundgates

P. Avouris et al., Nat. Mat., 605, 2, (2007)

F. Schwiertz, Nat. Nanotech. 5, 487 (2010)



Logic applications

I − Vg characteristics of a CNT-FET I = I(Vg ,Vds) GNR-FET



Band-gap opening

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



e-h symmetry

HTB =∑

σ,ij(tija†i,σbj,σ + tjib

†j,σai,σ)

Electron-hole symmetry

bi → −bi =⇒ h → −h

if εi is eigenvalue andc†i =

Pi αi a

†i +

Pj βj b

†j eigenvector

⇓

−εi is also eigenvalue andc

′†i =

Pi αi a

†i −

Pj βj b

†j is eigenvector

9>>>=>>>; n∗

nA + nB − 2n∗9>>>=>>>; n∗



Spatial symmetry

r-space

σd

σv

AC CB

C62I C

G0 = D6h⇒ G(K) = D3h

k-space

σd

C3

K

K K

G(k) = {g ∈ G0|gk = k + G} ⇒ G(K) = D3h



Spatial symmetry

r-space

σd

σv

AC CB

C62I C

G0 = D6h⇒ G(K) = D3h

k-space

C2v

D3h

D2h

sC

D6h

G(k) = {g ∈ G0|gk = k + G} ⇒ G(K) = D3h



Spatial symmetry

r-space

σd

σv

AC CB

C62I C

G0 = D6h⇒ G(K) = D3h

|Ak〉 = 1√NBK

PR∈BK e−ikR |AR〉

|Bk〉 = 1√NBK

PR∈BK e−ikR |BR〉

〈r |AR〉 = φpZ (r− R)

for k = K

{|Ak〉 , |Bk〉} span the E ′′ irrep of D3h



Spatial symmetry

Γ K Μ Γ

-10.0

-5.0

0.0

5.0

10.0

(ε−ε

F)

/ eV

NN only

TB fit to DFT data

D6h D3h

C2v

D2h

A2u B2

E"

B2g

A2

B2g

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

{|Ak〉 , |Bk〉} span the E ′′ irrep of D3h



Spatial and e-h symmetry

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 adegeneracy at the Fermi level, i.e. Egap = 0

Proof.Use electron-hole symmetry



A simple recipe

Consider nxn graphene superlattices (i.e. G = D6h): degeneracyis expected at Γ, KIntroduce pZ vacancies while preserving point symmetryCheck whether it is possible to turn the number of E irreps to beeven both at Γ and at K



Counting the number of E irreps

n = 4

K: EK: 2A + 2E : 2A: 2A + 2E ΓΓ

Γ A E0̄3 2m2 2m2

1̄3 2(3m2 + 2m + 1) 2(3m2 + 2m)

2̄3 2(3m2 + 4m + 2) 2(3m2 + 4m + 1)

Kn A E0̄3 2m2 2m2

1̄3 2m(3m + 2) 2m(3m + 2) + 12̄3 2(3m2 + 4m + 1) 2(3m2 + 4m + 1) + 1

⇒ n = 3m + 1, 3m + 2, m ∈ N



An example

(14, 0)-honeycomb



Band-gap opening..

Tight-binding

εgap(K ) ∼ 2t√

1.683/n

DFT

εgap(K ) ∼ 2t√

1.683/n

R. Martinazzo, S. Casolo and G.F. Tantardini, Phys. Rev. B, 81 245420 (2010)



..and Dirac cones

..not only: as degeneracy may still occur at ε 6= εFnew Dirac points are expected



Antidot superlattices

...the same holds for honeycomb antidots

1 nm

3 nm

0 10 20 30 40 50n

0.0

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 14

an / nm

0.0

0.1

0.2

0.3

0.4

0.5

ε gap /

eV



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)



..novel transistor?


Summary

Summary

Thermodynamically and kinetically favoured H clusters minimizesublattice imbalanceDefects can be used to open a gap without breaking the substratesymmetry


Summary

Acknowledgements

University of Milan

Gian Franco Tantardini

Simone Casolo

Matteo Bonfanti

Chemical Dynamics Theory Group

http://users.unimi.it/cdtg

University of OsloOle Martin LovvikISTMAlessandro Ponti

+-x:C.I.L.E.A. Supercomputing CenterNoturI.S.T.M.


Summary

Acknowledgements

Thank you for your attention!


defective graphene, not so bad after all

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