Structure of GrapheneChao Yin2019.10.8

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Outline

• Introduction of graphene• Structure of suspended graphene sheets [1]

• Mermin-Wagner theorem• experimental set up• identification of monolayer graphene• corrugation

• Moire pattern in multilayer graphene [2]• moire pattern• multilayer epitaxial graphene (MEG)• lattice orientation• strain

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[1] Nature 446, 60–63 (2007)[2] Phy. Rev. B. 295 (4) (1984)

Outline

• Introduction of graphene• Structure of suspended graphene sheets [1]

• Mermin-Wagner theorem• experimental set up• identification of monolayer graphene• corrugation

• Moire pattern in multilayer graphene [2]• moire pattern• multilayer epitaxial graphene (MEG)• lattice orientation• strain

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[1] Nature 446, 60–63 (2007)[2] Phy. Rev. B. 295 (4) (1984)

history

• 1947: theory for 3D graphite [1]• 1984: theory of Dirac fermion [2]• 1970s: single-layer epitaxial graphene,

electronic structure significantly altered by metal substrate

• A. Geim [3]:

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[1] Phy. Rev. 71 (9) (1947)[2] Phy. Rev. B. 295 (4) (1984)[3] Science. 324 (5934) (2009)

rediscovery

• 2010 Nobel Prize• production: micromechanical cleavage• SiO2: electrically isolated the graphene• gate voltage applied• Atomic force microscopy (AFM):

identify single layers

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Andre Geim Konstantin Novoselov

Science. 306 (5696) (2004)

properties

• electronic• massless Dirac fermions• high electron mobility and lowest resistivity at room temperature• Anomalous quantum Hall effect 𝜎𝜎𝑥𝑥𝑥𝑥 = ±4 𝑁𝑁 + 1

2𝑒𝑒2/ℎ

• Unconventional superconductivity in magic-angle graphene• structure: 2D crystal• chemical: hundred times more chemically

reactive than thicker sheets• optical: unexpectedly high opacity• mechanical: strongest material• … 6

applications

• already used [1]:• graphene-based touch panel modules• graphene tennis racquets• graphene-infused printer powder

• potential:• lightweight, thin and flexible

electric/photonics circuits• solar cells• medicine

7[1] en.wikipedia.org/wiki/Graphene

Outline

• Introduction of graphene• Structure of suspended graphene sheets [1]

• Mermin-Wagner theorem• experimental set up• identification of monolayer graphene• corrugation

• Moire pattern in multilayer graphene [2]• moire pattern• multilayer epitaxial graphene (MEG)• lattice orientation• strain

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[1] Nature 446, 60–63 (2007)[2] Phy. Rev. B. 295 (4) (1984)

Can a strictly 2D crystal exist?

• No!• theory: Mermin-Wagner theorem• experiment: decompose or segregate• previous graphene: supported by a bulk

substrate, not 2D• suspend it!

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Surf. Sci. Rep. 61, 1–128 (2006)

Mermin-Wagner theorem

• continuous symmetries cannot be spontaneously broken in dimensions 𝑑𝑑 ≤ 2

• (at 𝑇𝑇 > 0 in systems with sufficiently short-range interactions)• crystal: translational symmetry broken• discrete symmetry analog:

Ising model 𝛽𝛽𝛽𝛽 = 𝐾𝐾∑ 𝑖𝑖𝑖𝑖 𝜎𝜎𝑖𝑖𝜎𝜎𝑖𝑖

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Mermin-Wagner theorem• distortion field 𝒖𝒖(𝒙𝒙), Lame coefficients 𝜇𝜇, 𝜆𝜆

• energy 𝛽𝛽𝛽𝛽 = 12𝑉𝑉∑𝒒𝒒 𝜇𝜇𝑞𝑞2𝒖𝒖𝒒𝒒𝒖𝒖−𝒒𝒒 + 𝜇𝜇 + 𝜆𝜆 𝒒𝒒 ⋅ 𝒖𝒖𝒒𝒒 𝒒𝒒 ⋅ 𝒖𝒖−𝒒𝒒

• probability distribution 𝑃𝑃 𝒖𝒖𝒒𝒒 ∝ 𝑒𝑒−𝛽𝛽𝛽𝛽

• 𝑢𝑢𝑖𝑖,𝒒𝒒𝑢𝑢𝑖𝑖,𝒒𝒒′ ∝ 𝛿𝛿𝒒𝒒,−𝒒𝒒′/𝑞𝑞2

• 𝒖𝒖 𝒙𝒙 −𝒖𝒖 0 2 = ∑𝒒𝒒1,𝒒𝒒2 𝑢𝑢𝑖𝑖,𝒒𝒒1𝑒𝑒−𝑖𝑖𝒒𝒒1⋅𝒙𝒙 − 𝑢𝑢𝑖𝑖,𝒒𝒒1 𝑢𝑢𝑖𝑖,𝒒𝒒2𝑒𝑒

−𝑖𝑖𝒒𝒒2⋅𝒙𝒙 − 𝑢𝑢𝑖𝑖,𝒒𝒒2∝ ∫ 𝑑𝑑𝑑𝑑𝑞𝑞[2 − 2cos(𝒒𝒒 ⋅ 𝒙𝒙)]/𝑞𝑞2

∝ �𝒙𝒙 2−𝑑𝑑 , 𝑑𝑑 < 2

ln 𝒙𝒙 /𝑎𝑎 ,𝑑𝑑 = 2𝒙𝒙 0, 𝑑𝑑 > 2

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ocw.mit.edu/courses/physics 8.334

experimental set up

• micromechanical cleavage -> graphene on 𝑆𝑆𝑆𝑆𝑂𝑂2

• metal grid deposited on top by electron-beam lithography

• etch away the substrate

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identify monolayers

• electron diffraction

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corrugation

• evidence: peaks broader with increasing tilt angle

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phenomenology

• not strictly 2D

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quantitative

• cone angle -> direction ±5∘

• spatial extent 𝐿𝐿• lower limit: slightly smaller than 𝐿𝐿𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 10𝑛𝑛𝑛𝑛 of the diffracted electron

• upper limit 𝐿𝐿 ≤ 25𝑛𝑛𝑛𝑛: a large number of different orientations to provide Gaussian shape of peaks

• height in 3rd dimension ~1𝑛𝑛𝑛𝑛

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𝐿𝐿 numerics

• superimpose random sinusoidal waves𝐿𝐿 = 2𝑛𝑛𝑛𝑛 𝐿𝐿 = 5𝑛𝑛𝑛𝑛 𝐿𝐿 = 20𝑛𝑛𝑛𝑛

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TEM evidence• monolayer ו few-layer √• 𝐿𝐿 smaller

than monolayer

• static

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Outline

• Introduction of graphene• Structure of suspended graphene sheets [1]

• Mermin-Wagner theorem• experimental set up• identification of monolayer graphene• corrugation

• Moire pattern in multilayer graphene [2]• moire pattern• multilayer epitaxial graphene (MEG)• lattice orientation• strain

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[1] Nature 446, 60–63 (2007)[2] Phy. Rev. B. 295 (4) (1984)

moire pattern• origin: textile with a wavy

appearance produced mainly from silk

• optical moire interferometry: detect strain field

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Proc. Phys. Soc. London, Sect. B 69, 373 (1956)

multilayer epitaxial graphene (MEG)

• grown on 𝑆𝑆𝑆𝑆𝑆𝑆, ~10 layers• rotational stacking• monolayer electronic properties

21PRL 100, 125504 (2008)

lattice orientation• STM-based atomic moire interferometry• visibility decay with depth

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quantitative

• 1 layer image: 𝑓𝑓1 𝒃𝒃1,𝒃𝒃2;𝒙𝒙 = cos 𝒃𝒃1 ⋅ 𝒙𝒙 + cos 𝒃𝒃2 ⋅ 𝒙𝒙• 2 layers: 𝑓𝑓2 𝒙𝒙 = 𝑓𝑓1 𝒃𝒃1,𝒃𝒃2; 𝑥𝑥 + 𝑓𝑓1 𝒃𝒃1′ ,𝒃𝒃2′ ; 𝑥𝑥

= 2 cos (𝒃𝒃1 − 𝒃𝒃1′ ) ⋅ 𝒙𝒙/𝟐𝟐 cos (𝒃𝒃1 + 𝒃𝒃1′ ) ⋅ 𝒙𝒙/𝟐𝟐 + (1 ↔ 2)

• beat mode length 𝐴𝐴𝑖𝑖 = 2𝜋𝜋|𝒃𝒃𝑖𝑖−𝒃𝒃𝑖𝑖

′|

• 𝜃𝜃-rotation: 𝐷𝐷 = 𝐴𝐴1 = 𝐴𝐴2 = 𝑎𝑎2 sin(𝜃𝜃/2)

• 3 layers: 2 layers of morie patterns

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example

• FT -> 𝐷𝐷1 = 2.95 𝑛𝑛𝑛𝑛,𝐷𝐷2 = 3.35 𝑛𝑛𝑛𝑛 -> 𝜃𝜃1 = 4.78∘,𝜃𝜃2 = 4.21∘

• 𝐷𝐷3 = 𝐷𝐷1𝐷𝐷2

𝐷𝐷12+𝐷𝐷22−2𝐷𝐷1𝐷𝐷2 cos ∆𝜙𝜙= 24.7 ± 1.6 𝑛𝑛𝑛𝑛

• from STM: 𝐷𝐷3 ≈ 26 𝑛𝑛𝑛𝑛

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further proof

• change sample bias at constant tunneling current

• 2 moire patterns at different height

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strain

• mechanisms• boundary• unequal thermal

contractions between SiCand graphene

• uniform relative strain in the arm chair direction

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𝐷𝐷𝑚𝑚𝑎𝑎𝑥𝑥/𝐷𝐷𝑚𝑚𝑖𝑖𝑐𝑐 = 1.23, 𝜖𝜖 ≈ 0.37%

Conclusion

• Graphene electronic properties• Structure of suspended graphene sheets

• corrugation stabilizes 2D crystals• data analysis

• Moire pattern in multilayer graphene• moire pattern as a microscope

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Conclusion

• Graphene electronic properties• Structure of suspended graphene sheets

• corrugation stabilizes 2D crystals• data analysis

• Moire pattern in multilayer graphene• moire pattern as a microscope

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questions?