Graphing and Grafting Graphene: Classifying Finite Topological Defects(PRB 83, 195425 (2011); arXiv:1106.6273)

Eric Cockayne, Joseph Stroscio, Gregory Rutter, Nathan Guisinger, Jason Crain

NIST

Phillip FirstGeorgia Tech

Subject of the 2010 Nobel prize in Physics

Castro-Neto et al., Physics World (2006)

Castro-Neto, Nature Mater. 6, 176 (2007).

Graphene:

Unusual electronic structure makes it a promising candidatefor applications

Microelectronics: high carrier mobility → high speed devices

Resistance standard → unusual quantum Hall effect

Commercial applications will require methods for large-scale production

Graphene production methodsMechanical exfoliation from graphiteChemical exfoliation from graphiteChemical reduction of graphene oxideSegregation of carbon from metal carbidesChemical vapor deposition of C onto metal surfacesThermal desorption of Si from SiC

Growth of graphene from by thermaldesorption of Si from SiC very promising,but defects frequently observed

Goal of this talk: elucidate nature of defectswith the ultimate aim of (1) reducing oreliminating the defects or (2) generating defectsat will to tune properties.

Key results of this work:

Topological defects are among the most common.

Systematic way of describing and identifying topological defects found.

Topological defects in graphene:

Changing the number of sides in a ring (replacing hexagons with pentagons, heptagons, etc.sp2 bonding: C planar; 3 neighborsAverage number of sides = 6 exactly

Average < 6; positive curvature; buckyballsAverage >6; negative curvature: “schwartize”

Keep” flat”: defect with more than 6 membered ring mustbe compensated with ring with < 6 members & vice versa

R. Phillips et al,PRB 46, 1941(1992).

Positive and negative disclinations: parents of all topologicaldefects in graphene

Combine one positive and one negative disclination:obtain dislocation core.

Linear array of dislocation cores: grain boundary

Grain boundary that closes on itself:

Grain boundary loop

The grain boundary loop is the first type of topological defect shown in this talk that is “local”

Local topological defect: core region of defect surrounded by lattice that is topologically equivalent to defect-free graphene

Because only the core region is “disturbed”, these defects might be created or annihilated by the rearrangement of relatively few C atoms

May be among most important defects in graphene

Hypothesis: defects seen in earlier STM images are local topological defects.

“Flower” defect

Ab initio electronic structure

VASP used DFT, ultrasoft pseudopotentials212 eV plane wave cutoff; Up to 864 atoms in supercell8748 k points per BZ of primitive cell

STM topographs simulations

Tersoff approximation: Fixed V Current proportional to local density of states between Fermi level and bias V

Tight binding model

C 2pz levels put into model (Tanaka et. al., Carbon)Parameters determined via least squares fitting to ab initio dataUp to 3888 atoms for bilayer supercell

Computational Methods

Simulated STM images of the “flower” defect matches experiment.

Cockayne et al., PRB 83, 195425 (2011).

dI/dV plot ~ local density of states sharply peaked in energy,about 0.5 eV above the Dirac point

Scanning tunneling spectroscopy of defects: energy-resolved information

Experimental dI/dE plots compared w/ computed density of states

Three computed peaks in experimental range.

Why is only one seen?

Calculated DOS corresponding to each peak:

Tight binding model confirms that peaks 2 and 3 come from single resonance inside flower region at ED + 0.3 eV and suggests that peak 4 is weak, explaining experimental observation of single peak

The wavefunction of peak 1 is clearly different from the rest.

Although peak 4 looks superficially similar to the resonance of peaks 2 and 3, tight binding calculations show that it has a different symmetry.

Energetic/Mechanical properties of flower defect:

Lowest energy per 5-7 pair of any known topological defect

Likely to coalesce mobile dislocation cores if they can notheal out.

Also has large A/E:

May increase strength of graphene under isotropic tension

Ideal graphene Cut Rotate

Paste

“Flower” defect:

Equivalent to rotating aportion of graphene with respect to the rest:

A variety of rotational grain boundaries existwith different symmetries, number of core atoms rotated, and rotation angle

Under hexagonal symmetry, there exists a whole family of rotational grain boundaries

Labeled by pair of integers (m,n)

Central 6 m2 atoms rotated by 60o (n/m)

(2,1) (3,1) (4,1)

(4,2)

It is also possible to create a grain boundary loop by cutting out aregion and then splicing a region with a different number of C atoms.

As long as the number of “dangling bonds” is equal, the threefold bonding requirement will be satisfied.

This allows for divacancies and di-interstitials to reconstruct and lower energy

Need systematic way to classify “grain boundary loop” defects

Key to systematic classification:describe graphene defect structures in terms of the dual lattice.

Dual lattice: n-vertices go to n-tiles and vice versa.Dual lattice of the graphene honeycomb structure is triangular.sp2 bonding (all vertices 3-vertices) means that all dual structures consist of triangles only

Work in “dual space”

Don’t design defective graphene structures. Design defective triangle structures, and then take dual.

In analogy with Stone-Wales defect, one can take any patch of triangles, and retriangulate in a way that preserves the perimeter. (examples will be shown in next slides).

Structures will presumably have low energy if the retriangulation is also a patch of the ideal triangular lattice.

Then, in terms of the graphene structure, one cuts out a patch and then “splices” or “transplants” a different patch of graphene with the same number of dangling bonds.

Interesting mathematical result follows: topological defects that preserve sp2 bonding can only keep the number of atoms identical or change it by a multiple of 2.

Rule of thumb: topological defects in graphene prefer to have alternating 5-rings and 7-rings.

Above image: graphene grain bounday (Huang et al., Nature (2011)).

One can design a defect with alternating 5-rings and 7-rings in dual space by choosing a replacement patch in dual space where the vertices have alternately +1 and -1 the number of triangles of the original.

Metaphorically, one looks for “most compatible donors”

The complete set of “most-compatible donor” topological defects (with constraints on size and change in number of atoms <=2) is shown on right

An infinite number of larger defects exists.

The graphical representation of each defect is to draw an arrow connecting the original patch of triangles in the dual (opposite side) to the replacment patch (same side).

The topological defects occur as inverse pairs.

Shown are change in number of atoms (top) and DFT formation energy (in eV, bottom).

ab

c d

ef

gh

Simulated STM images of small topological defects.

J. Kotakoski et al., Phys. Rev. Lett. 106 105505 (2011)

In addition to exploring defects systematically, the above paper suggests looking at defect clusters in addition to isolated local defects.

Previously unidentified experimental defects identified as isolated divacancies (top right) or divacancy clusters(double divacancy, bottom left)), (triple divacancy (bottom right)

The triple divacancy has a triangle of carbon atoms in the center!

Is this right?

Triple divacancy Experiment Single vacancy

Conclusions

Dual space gives a graphical representation for finite topological defects in graphene

Low energy defects can be designed via “most compatible donor” procedure

Previously unidentified defects in graphene are identified as divacancies or divacany clusters

One such defect contains a planar triangle of carbons, an unusual structural motif

Other Graphene defects:

Impurity atom

Substitution

Defect atom can be anything: focus on Mo and Si

Best fit: intercalated Mo

Adatom Intercalation Substitution

Defect atom can be anything

Graphene layers remain nearly flat (h < 0.25 A) for intercalated Mo

Magnetism?Mo position Magnetic moment M(B)

isolated atom 6.0adatom 0.0intercalated 0.0substitution 2.0 c/w M = 2.0 for Cr substitution in monolayer (Krasheninnikov et al., PRL 102, 126807 (2009); Santos et al,arXiv:0910.0400)

324 C + 1 Mo: DOS shows three defect-associated peaks near EDirac

Plots a-g show, inorder of increasingenergy, the zone centerstates near EDirac withsignificant Mo d participation

State E-EDirac mult. a -0.57 2 b -0.49 1 c -0.17 2 d +0.02 2 e +0.20 1 f +0.33 2 g +0.88 2

Range of E ~ 1.5 eVBright center: singlet; m = 0; dark center; doublet; m nonzero

Can individual state(s) be identified that match experimental STM images?