raman spectroscopy of graphene - fb physik, fu berlin · 2014-01-15 · stefan ulonska (fu berlin)...
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Raman spectroscopy of graphene
Stefan Ulonska
Free University of Berlin
01/14/2014
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 1 / 27
Outline
1 Raman scatteringMicroscopic description of Raman scatteringResonant Raman scatteringHigher order scatteringSymmetry selection rules
2 Raman spectroscopy of grapheneRaman spectrum of grapheneCharacterization of graphene
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 2 / 27
Outline
1 Raman scatteringMicroscopic description of Raman scatteringResonant Raman scatteringHigher order scatteringSymmetry selection rules
2 Raman spectroscopy of graphene
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 3 / 27
Visible light scattering on matter
Conservation requirements:
~ωsc = ~ωi ± ~Ωph
~ksc = ~ki ± ~qph
ωsc = ωi : Rayleighωsc = ωi − Ωph: Stokesωsc = ωi + Ωph: Anti-Stokes
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 4 / 27
Fundamental Raman selection rule
Considerations for ~qph
Maximal magnitude |~q| = 2 · |~ki | for backscattering.
Visible light at 530 nm: O(|~ki |)= 10−3 Å−1
Phonon Brillouin zone(BZ) edge: O(|~qph|)=πa = Å−1
[1]
|~qmax | = 2 · |~ki | ≈ 0 at Γ-pointof phonon BZ
At Γ: only optical phonons withΩph(q = 0) 6= 0 can contributeto first order Raman signal
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 5 / 27
Microscopic picture of Stokes process
Quantum state:|photon,phonon,electron〉Electronic: |i〉 ⇒ |n〉 ⇒ |n∗〉 ⇒ |i〉
Individual transition rates(1) |ωi ,0, i〉 ⇒ |0,0,n〉 :
k1 ∝∑
n〈0,0,n |Hσ |ωi ,0,i〉
~ωi−(En−Ei )
(2) |0,0,n〉 ⇒ |0,ph,n∗〉 :
k2 ∝∑
n,n∗〈0,ph,n∗ |He−ph | 0,0,n〉~ωi−~Ωph−(En∗−En)
(3) |0,ph,n∗〉 ⇒ |ωsc ,ph, i〉 :
k3 ∝∑
n,n∗〈ωsc ,ph,i |Hρ | 0,ph,n∗〉
~ωi−~Ωph−~ωsc
Electronic excitation mediatesRaman scattering of photonsElectron remains unchanged afterscattering:Same final state |i〉 as before photonabsorption
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 6 / 27
Microscopic picture of Stokes process
Quantum state:|photon,phonon,electron〉Electronic: |i〉 ⇒ |n〉 ⇒ |n∗〉 ⇒ |i〉
Individual transition rates(1) |ωi ,0, i〉 ⇒ |0,0,n〉 :
k1 ∝∑
n〈0,0,n |Hσ |ωi ,0,i〉
~ωi−(En−Ei )
(2) |0,0,n〉 ⇒ |0,ph,n∗〉 :
k2 ∝∑
n,n∗〈0,ph,n∗ |He−ph | 0,0,n〉~ωi−~Ωph−(En∗−En)
(3) |0,ph,n∗〉 ⇒ |ωsc ,ph, i〉 :
k3 ∝∑
n,n∗〈ωsc ,ph,i |Hρ | 0,ph,n∗〉
~ωi−~Ωph−~ωsc
Electronic excitation mediatesRaman scattering of photonsElectron remains unchanged afterscattering:Same final state |i〉 as before photonabsorption
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 6 / 27
Microscopic picture of Stokes process
Quantum state:|photon,phonon,electron〉Electronic: |i〉 ⇒ |n〉 ⇒ |n∗〉 ⇒ |i〉
Individual transition rates(1) |ωi ,0, i〉 ⇒ |0,0,n〉 :
k1 ∝∑
n〈0,0,n |Hσ |ωi ,0,i〉
~ωi−(En−Ei )
(2) |0,0,n〉 ⇒ |0,ph,n∗〉 :
k2 ∝∑
n,n∗〈0,ph,n∗ |He−ph | 0,0,n〉~ωi−~Ωph−(En∗−En)
(3) |0,ph,n∗〉 ⇒ |ωsc ,ph, i〉 :
k3 ∝∑
n,n∗〈ωsc ,ph,i |Hρ | 0,ph,n∗〉
~ωi−~Ωph−~ωsc
Electronic excitation mediatesRaman scattering of photonsElectron remains unchanged afterscattering:Same final state |i〉 as before photonabsorption
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 6 / 27
Transition probability of first order Raman scattering
Neglecting other time orders[2], the Raman scattering probability is given byFermi’s rule:
P =2π~· |k1 · k2 · k3|2
Total transition probability
P =2π~· |∑n,n∗
〈i |Hρ |n∗〉 · 〈n∗ |He−ph |n〉 · 〈n |Hσ | i〉[~ωi − (En − Ei )] · [~ωi − ~Ωph − (En∗ − Ei )]
|2 · δ(ωi − Ωph − ωsc)
• Matrix elements depend on symmetries of Hρ,σ,e−ph and |i〉 , |n〉 , |n∗〉⇒ Symmetry selection rules
• Vanishing (or small) denominators increase scattering probability⇒ Resonant Raman scattering
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 7 / 27
Transition probability of first order Raman scattering
Neglecting other time orders[2], the Raman scattering probability is given byFermi’s rule:
P =2π~· |k1 · k2 · k3|2
Total transition probability
P =2π~· |∑n,n∗
〈i |Hρ |n∗〉 · 〈n∗ |He−ph |n〉 · 〈n |Hσ | i〉[~ωi − (En − Ei )] · [~ωi − ~Ωph − (En∗ − Ei )]
|2 · δ(ωi − Ωph − ωsc)
• Matrix elements depend on symmetries of Hρ,σ,e−ph and |i〉 , |n〉 , |n∗〉⇒ Symmetry selection rules
• Vanishing (or small) denominators increase scattering probability⇒ Resonant Raman scattering
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 7 / 27
Resonant Raman scattering
Resonant transition probability at critical points[2]
Pres ≈2π~· |〈i |Hρ |n〉 · 〈n |He−ph |n〉 · 〈n |Hσ | i〉
[~ωi − En] · [~ωsc − En]+ C|2·
2 resonant processes~ωi = En: Incoming resonance~ωsc = En: Outgoing resonance
• Resonant Raman scattering can be usedto determine Ωph and critical points of e−
bandstructure• Scattering signal enhancement≈ 105 − 107
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 8 / 27
Resonant Raman scattering
Resonant transition probability at critical points[2]
Pres ≈2π~· |〈i |Hρ |n〉 · 〈n |He−ph |n〉 · 〈n |Hσ | i〉
[~ωi − En] · [~ωsc − En]+ C|2·
2 resonant processes~ωi = En: Incoming resonance~ωsc = En: Outgoing resonance
• Resonant Raman scattering can be usedto determine Ωph and critical points of e−
bandstructure• Scattering signal enhancement≈ 105 − 107
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 8 / 27
Higher order Raman scattering
Two-phonon scatteringMomentum conservation:∑
i ~qi ≈ 0Possible by exciting thesame(overtone) or differentphonon branches(combination)
⇒ Allows for study of phononDOS in some materials
Other higher order processesElectronic scattering on defects:⇒ Important in graphene to study sample structure and quality
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 9 / 27
Higher order Raman scattering
Two-phonon scatteringMomentum conservation:∑
i ~qi ≈ 0Possible by exciting thesame(overtone) or differentphonon branches(combination)
⇒ Allows for study of phononDOS in some materials
Other higher order processesElectronic scattering on defects:⇒ Important in graphene to study sample structure and quality
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 9 / 27
Raman selection rules
Result of group theory• All possible functions (electrons, phonons, physical properties) in a
crystal transform uniquely under symmetry operations⇒ A defined symmetry (e.g. A1g , B2u) can be assigned to each state inthe crystal⇒ Symmetry properties of a crystal and its functions are summarized incharacter tables
Symmetry of matrix elements
M = 〈i |Hρ |n∗〉 · 〈n∗ |He−ph |n〉 · 〈n |Hσ | i〉
• M 6= 0 only if Γ(Hσ)⊗ Γ(Hρ)⊗ Γ(He−ph) 6= 0• Dipole approximation: Hσ and Hρ have the same symmetry as the
polarization coordinates of the photons.• He−ph transforms like the phonon with symmetry Γph
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 10 / 27
Raman selection rules
Result of group theory• All possible functions (electrons, phonons, physical properties) in a
crystal transform uniquely under symmetry operations⇒ A defined symmetry (e.g. A1g , B2u) can be assigned to each state inthe crystal⇒ Symmetry properties of a crystal and its functions are summarized incharacter tables
Symmetry of matrix elements
M = 〈i |Hρ |n∗〉 · 〈n∗ |He−ph |n〉 · 〈n |Hσ | i〉
• M 6= 0 only if Γ(Hσ)⊗ Γ(Hρ)⊗ Γ(He−ph) 6= 0• Dipole approximation: Hσ and Hρ have the same symmetry as the
polarization coordinates of the photons.• He−ph transforms like the phonon with symmetry Γph
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 10 / 27
Raman selection rules
Symmetry conditions on matrix elements
Γ(Hσ)⊗ Γ(Hρ) ⊃ Γph
Excited phonon has to have the same symmetry as the product of therespective polarization coordinates of Hρ and Hσ
⇒ Determination of phonon symmetry via polarization dependent Ramansignal
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 11 / 27
Raman selection rules
Symmetry conditions on matrix elements
Γ(Hσ)⊗ Γ(Hρ) ⊃ Γph
Excited phonon has to have the same symmetry as the product of therespective polarization coordinates of Hρ and Hσ
⇒ Determination of phonon symmetry via polarization dependent Ramansignal
Example: graphene, D6h group
[3]
Hσ, Hρ along ~x direction (polarizers)⇒ Phonon with A1g symmetry could be measured since it transforms like x2
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 12 / 27
Outline
1 Raman scattering
2 Raman spectroscopy of grapheneRaman spectrum of grapheneCharacterization of graphene
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 13 / 27
Graphene recap
[4]
Crystal properties• 2-dimensional material with hexagonal symmetry (D6h space group)• 2 carbon atoms per unit cell
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 14 / 27
Phonon dispersion of graphene
[1] [1]
6 phonon branches• 4 different
symmetries
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 15 / 27
D6h character table
[3]
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 16 / 27
D6h character table
[3]
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 17 / 27
Raman spectrum of graphene
[4]
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 18 / 27
Raman spectrum of graphene
[4]
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 19 / 27
Raman D peak in graphene
[4]
D• Intervalley scattering with TO
phonon at K-point, ~qph 6= 0• D peak requires defect
scattering to conservemomentum
2D• Double-resonant, intervalley
scattering[5] requires nodefects
• Strong e−-ph coupling: 2Dpeak is more intense than G
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 20 / 27
Determination of defects and number of layers
[6]
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 21 / 27
Determination of defects and number of layers
Seminar paper[6]"Position and fwhm of the G and 2D peak(...)confirm presence of single-layergraphene"
[7]
• Position of 2D peak dependson number of layers
• Additional layers increaseFWHM of 2D peak (more e−
states for resonant scattering)• Occurence of peaks at 1350
cm−1 (D) and 1620 cm−1 (D’)reveal defects of graphenesample
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 22 / 27
Determination of defects and number of layers
Seminar paper[6]"Position and fwhm of the G and 2D peak(...)confirm presence of single-layergraphene"
[7]
• Position of 2D peak dependson number of layers
• Additional layers increaseFWHM of 2D peak (more e−
states for resonant scattering)• Occurence of peaks at 1350
cm−1 (D) and 1620 cm−1 (D’)reveal defects of graphenesample
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 22 / 27
Interference with substrate
Seminar paper[6]"This is supported by a peak height ratio 2D/G of 2.8 which [indicates](...)anoxide layer of 300 nm thickness"
[8][8]
Interference effects of Raman signal and substrate allow for determination oflayer thickness from peak ratio 2D/G
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 23 / 27
Determination of strain components
[6]
Phonon frequencies change under strain[9]
∆ω± = ∆ωh ± 12
∆ωS
= −ω0 · γ · εh ±12ω0β · εS
• Hydrostatic strain ∆ωh shifts phonon frequency• Degenerate phonon modes split due to shear strain ∆ωS
• Frequency shift allows for determination of strain tensor ¯ε
⇒ Raman signals sensitive to sample strain and relative orientation
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 24 / 27
Determination of strain components
[6]
Phonon frequencies change under strain[9]
∆ω± = ∆ωh ± 12
∆ωS
= −ω0 · γ · εh ±12ω0β · εS
• Hydrostatic strain ∆ωh shifts phonon frequency• Degenerate phonon modes split due to shear strain ∆ωS
• Frequency shift allows for determination of strain tensor ¯ε
⇒ Raman signals sensitive to sample strain and relative orientation
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 24 / 27
Determination of strain components
[6]
Seminar paper[6]
“In areas of plasmonic enhancementthe graphene is under strain with ahydrostatic component ≈0.8% and ashear component <0.4%"
• Frequency shift of peaks on Aunanostructure
• Modes broadened but not split inareas of strain
⇒ Calculate hydrostatic strain,approximate shear strain
⇒ Plasmonic signal enhancement inareas under strain
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 25 / 27
Determination of strain components
[6]
Seminar paper[6]
“In areas of plasmonic enhancementthe graphene is under strain with ahydrostatic component ≈0.8% and ashear component <0.4%"
• Frequency shift of peaks on Aunanostructure
• Modes broadened but not split inareas of strain
⇒ Calculate hydrostatic strain,approximate shear strain
⇒ Plasmonic signal enhancement inareas under strain
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 25 / 27
Summary
Raman scattering• Non-destructive technique to determine both electronic and vibrational
properties• Identification of phonon symmetry and energy• Allows study of samples with both spatial and frequency resolution
Raman spectroscopy of graphene• Identification of defects, substrate thickness and number of layers• Resolution of regions under strain and determination of strain
components• Study electron-phonon and electron-electron interactions• (...)
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 26 / 27
Summary
Raman scattering• Non-destructive technique to determine both electronic and vibrational
properties• Identification of phonon symmetry and energy• Allows study of samples with both spatial and frequency resolution
Raman spectroscopy of graphene• Identification of defects, substrate thickness and number of layers• Resolution of regions under strain and determination of strain
components• Study electron-phonon and electron-electron interactions• (...)
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 26 / 27
References
[1] Popov. Non-adiabatic phonon dispersion of graphene. BJP, 2011.
[2] Yu Cardona. Fundamentals of Semiconductors: Physics and Materials Properties. 2005.
[3] webqc.org. D6h - point group symmetry character tables. URLhttp://www.webqc.org/symmetrypointgroup-d6h.html.
[4] Ferrari. Raman spectroscopy as a versatile tool for studying the properties of graphene.Nature Nanotechnology, 2013.
[5] Reich. Double resonant raman scattering in graphite. PRL, 2000.
[6] Heeg. Polarized plasmonic enhancement by au nanostructures probed through ramanscattering of suspended graphene. Nano Letters, 2012.
[7] Ferrari. Raman spectrum of graphene and graphene layers. PRL, 2006.
[8] Yoon. Interference effect on raman spectrum of graphene on sio2 /si. PRB, 2009.
[9] Mohiuddin. Uniaxial strain in graphene by raman spectroscopy: G peak splitting, grüneisenparameters, and sample orientation. PRB, 2009.
Stefan Ulonska (FU Berlin) Raman spectroscopy of graphene 01/14/2014 27 / 27