Landau level spectroscopy of graphene (Raman scattering and - - PowerPoint PPT Presentation

Landau level spectroscopy of graphene

(Raman scattering and far-infrared absorption)

Electron-phonon and electron-electron interactions

Marek Potemski

Laboratoire National des Champs Magnétiques Intenses Grenoble High Magnetic Field Laboratory CNRS/UJF/UPS/INSA

MOMB

PRL 114, 126804, (2015) "The ZOO of magneto-phonon resonances in graphene" D.M. Basko, P. Leszczynski, C. Faugeras… et al., to be published

Why ? Graphene: a truly two-dimensional crystal of sp2 –bonded carbon Graphene-Based Revolutions in ICT And Beyond This talk: fundamental properties studied with magnetic fields (spectroscopy)

= B > B

) (k E E  = ) (B E E

i

n

=

!

Dispersion relations and corresponding Landau level ladders

Electronic states, generic (quasi) 2D structure of sp2 carbon (Bernal stacking)

~ graphene + (effective) bilayers

= B > B

) (k E E  = ) (B E E

i

n

=

?!

Dispersion relations and corresponding Landau level ladders ~ graphene + (effective) bilayers Electronic states, generic (quasi) 2D structure of sp2 carbon (Bernal stacking)

Landau level spectroscopy

E B

Probing inter Landau level excitations :

j i

L L →

j

L

j

L

Absorption/transmission ) , ( B h T T ν =

− + σ

σ ν , h B resonant when

exc

E h = ν

E

+

+ = ∆ σ : 1 n

−

− = ∆ σ : 1 n

B

+

= 4 n = n

−

=1 n

+

= 3 n

+

= 2 n

+

=1 n

−

= 2 n

−

= 3 n

−

= 4 n Selection rules 1 ± = ∆ n 8 , 7 , 5 , 4 , 2 = ∆ n and if trigonal warping

V.P. Gusynin & S.G. Sharapov, PRB, 2006

• M. Koshino & T. Ando, PRB, 2008
• M. Mucha-Kruczynski et al., J. Phys., 2009

M.L. Sadowski et al., SSC, 2007

Raman scattering

− + σ

σ /

B ν h ν ν h h Eexc − = ' ' ν h

− + σ

σ /

Selection rules

E B

+

= 4 n = n

−

=1 n

+

= 3 n

+

= 2 n

+

=1 n

−

= 2 n

−

= 3 n

−

= 4 n = ∆ n

− − + +

σ σ σ σ / , /

strong 2 ± = ∆ n

+ − − +

σ σ σ σ / , /

weaker 1 ± = ∆ n if trigonal warping

• r coupled to phonon
• O. Kashuba & V.I. Falko PRB, 2009; M. Mucha-Kruczynski et al., PRB, 2010

excitation miniaturized optical bench collection

• ptical fibers

T B 30 K T 1

x,y,z-stage

Faugeras, Kossacki, Breslavetz, …

What can be learned from magneto-optics ? Band structure Scattering: efficiency ( mechanism ) ? √

Scattering ? Spectral broadening Classical condition for observation of cyclotron resonance (Landau quantization)

cyclotron scattering

T > τ

C S

ω τ / 1 >

min

/ 1 B > µ Scattering mechanisms More general :

scat

τ / 1 ← Γ ) , ( E B Γ = Γ ←

rough estimate of carrier mobility B  × e

What can be learned ? Band structure Scattering: efficiency ( and mechanism ) Interactions (?) : electron-phonon electron-electron √ √

E B

m n

L L → E

Interactions ? tuning the excitations in resonance

Interactions ? resonant electron-phonon coupling ?

E B

phonon

E

m n

L L →

δ

δ ← strength of interaction + more than this !!

Eph = E1,3 n = 0 1

• 2
• 3
• 1

3 2

• ther magneto-phonon" resonance

Esingle particle k ~ le-h/lB

2

E

C

ω  Eopt = Esingle particle =

B

l e ε π

2

2 2 1

= Eexch Restoring single electron spectrum

• f excitations at k ~ 0 (Kohn theorem)

e.g., C. Kalin, B.I. Halperin, PRB , 1984, Bychkov, Eliashberg, Iordanskii, (JETP Letters, 1981)

n = 0 n = 2 n = 1 Parabolic dispersions equidistant LLs Electron-electron interactions and inter Landau level transitions

kopt ~ 1/λ << kcoll ~1/lB

Optics is useless to study the many-body effects !?

Esingle particle k ~ le-h/lB

2

E

corr

∆

exch C

E E ≠

Eopt = Esingle particle +

? ∞ ÷ = ∆ δ

corr

n = -1 n = 0 n = 2 n = 1 n = -2

Expectations: Rather large deviations from effective single electron model ?

• A. Iyengar, et al., PRB, 2008 Yu.A. Bychkov, G. Martinez, PRB, 2008

Linear dispersions non-equidistant spacing Electron-electron interactions and inter Landau level transitions

• R. Roldan et al., PRB, 2010
• J. Sari, C. Toke, PRB, 2013

? ~ B

nm corr

γ ∆

YU.E. Lozovik, A.A. Sokolik, Nanoscale Research Lett., 2012

Graphene: Electron-electron interactions at B=0

OUTLINE Band structure mono to pentalayer graphene Scattering efficiency graphene on graphite: the best ever seen graphene Electron-phonon interaction the ZOO of magneto-phonon resonances Electron-electron interaction Conclusions

What can be learned from magneto-optics ? Band structure !

What can be learned from magneto-optics ? Band structure !

What can be learned from magneto-optics ? Band structure !

What can be learned from magneto-optics ? Band structure !

What can be learned from magneto-optics ? Scattering: efficiency ( mechanism ) !

• G. Li et al., PPRL, 2008

Graphene on graphite: best ever seen graphene !!

− 1

L L

+ 1

L

+ 2

L

+ 3

L

− 2

L

. . .

Cyclotron resonance absorption : high temperature but well resolved LLs

F

E E

LL spacing > kT LL broadening < LL spacing non-equidistant spacing multimode cyclotron resonance absorption

(Very) low field cyclotron resonance absorption Graphene on graphite mT 1 s m vF

6

10 . 1 ⋅ = perfect Dirac states :

LL broadening :

) 4 . ( 35 K eV µ ≈ Γ

• P. Neugebauer et. al., PRL, 2009

How perfect can graphene be Landau level quantization down to B0 = 1 mT

s V cm mT ⋅ = >

2 7

10 1 1 µ ) 4 . ( 35 K eV µ γ = m l s V cm m e ps

F

µ τ µ τ 20 , 10 3 , 20

2 7

≈ ⋅ ⋅ ≈ = ≈

∗ e F F F

m v E m cm n meV E

3 2 2 9

10 3 . 1 / , 10 3 , 5 . 6

− ∗ −

⋅ ≅ = ⋅ ≅ ≅

1

E = γ T B µ 1 = T BEarth µ 50 ≈ K K meV E 4 . 3 25 .

1

= > = ≈ γ Also at 50 K !

Pronounced Landau quantization in the magnetic field of the Earth

• P. Neugebauer et. al., PRL, 2009

Graphene on graphite: magneto Raman scattering response phonons + search for a characteristic electronic response e.g., L-1 → L1 inter Landau level excitation

1400 1600 1800 2000 2200 2400 2600 2800 500 1000 1500

L-1,1

Intensity (counts) Raman Shift (cm

• 1)

L-1,2/L-2,1 2D band G band

B=10T λ

exc.=514.53nm

Temp.=4K 25 50 75 100 125 150 25 50 75 100 125 150 25 50 75 100 125 150 25 50 75 100 125 150 25 50 75 100 125 150 25 50 75 100 125 150 25 50 75 100 125 150 25 50 75 100 125 150

d) c) b)

Distance (µm)

a)

Distance (µm) Distance (µm) Distance (µm) Distance (µm) Distance (µm)

E

B

L-1 → L1 2D band

Graphene on graphite: magneto-Raman scattering response: an overview

1500 2000 2500 3000 3500 Intensity (counts) Raman shift (cm

• 1)

B=0T B= 6T 2D G 2D’ 2D’’

1000 1200 1400 1600 1800 2000 2200 2400 2600 200 400 600 800 1000 1200 B=5.98T B=4.98T B=4.38T B=3.88T Intensity (counts) Raman shift (cm

• 1)

B=0T

Graphene on graphite: magneto-Raman scattering response: an overview

• C. Faugeras et al., PRL, 2011; M. Kühne et al., PRB, 2012, P. Leszczynski et. al, to be published

E B

+

= 4 n = n

−

=1 n

+

= 3 n

+

= 2 n

+

=1 n

−

= 2 n

−

= 3 n

−

= 4 n

E2g phonon + electonic excitations

Graphene on graphite: magneto-Raman scattering response: an overview

• C. Faugeras et al., PRL, 2011; M. Kühne et al., PRB, 2012, D. Basko et. al, to be published

focus on E2g phonon

Interactions ? resonant electron-phonon coupling ! E B

phonon

E

m n

L L →

δ

δ ← strength of interaction

In magnetic fields Resonant coupling of E2g phonon ("optical") with Δn=±1 inter Landau level excitations Theoretical predictions :

i f res i f res

f f B f f B E ) 1 ( ~ ) 1 ( ) ( 2 ~

1

− ⋅ ⋅ − ⋅ ⋅ λ λ δ

• T. Ando, JPSJ, 2007

M.O. Goerbig, et al., PRL , 2007

δ

− + σ

σ /

+ − σ

σ /

Magneto-phonon resonance: graphene on graphite

• C. Faugeras, et al., PRL, 2011; M. Kühne el al., PRB 2012
• J. Yan et al., PRL , 2010

Graphene on graphite: an electronic system of unprecedented quality !

Magnetic field (T)

5 10

Raman shift (cm-1)

1500 1600

1 2 3 4 5 1570 1580 1590 1600 1610 1620 1630

Raman shift (cm-1) B1/2 (T1/2)

T1 T2 T3 T4 Experiment: magneto-phonon resonance in epitaxial graphene

E 1 T2 T3

B E EF Neutral graphene

3

10 5 . 4

−

⋅ = λ

• C. Faugeras, et al., PRL, 2009

Magneto-phonon resonance in doped graphene Graphene flake on Si/SiO2

• P. Kossacki et al., Phys. Rev. B, 2012

i f res

f f B ) 1 ( ~ − ⋅ ⋅ λ δ

1500 1600 1700 5 10 15 20 25 30

B (T) Raman shift (cm-1)

1500 1600 1700 5 10 15 20 25 30

B (T) Raman shift (cm-1)

Experiment in qualitative agreement with simulations

• P. Leszczynski, A. Nicolet, C. Faugeras et al., to be published

Magneto-phonon resonances: graphene on h-BN ~ neutral and better electronic quality

• A. Nicolet, Y. Henni, C. Faugeras et al., to be published

Experiment in qualitative agreement with simulations Magneto-phonon resonances: bilayer graphene on h-BN

2 12 3 6

10 5 . 2 , 100 10 5 . 3 , / 10 06 . 1

− −

⋅ < < ⋅ = ⋅ = cm n meV E s m v

F F

λ

5 10 15 20 25 1500 1550 1600 1650

Raman shift (cm

• 1)

B (T)

T=4.2 K σ+/σ-

Phonon coupling to Δn=±1 inter Landau band transitions from the vicinity of the K-point + of the H point Magneto-phonon resonance in graphite

• P. Kossacki et al., PRB, 2011

5 10 15 20 25 30 B (T)

1580 1584 1588 1592

5 10 15 20 25 30

10 20 30

Raman shift (cm-1) FWHM (cm-1) B (T)

10 20 30 500 1000 1500 2000

kZ=2π/5 kZ=π/5

Raman shift (cm-1) B (T)

Probing the band structure with magneto-phonon resonance few (?) layer graphene on Si/SiO2 weakly doped tetra-layer graphene !

• C. Faugeras et al., New J. Phys., 2012

Graphene on graphite: magneto-Raman scattering Graphene on graphite: magneto-Raman scattering response: electronic excitationsan

5 10 15 20 25 30 1000 2000 3000 4000 5000 L-5,5L-5,4(-4,5) L-4,4 L-4,3 (-3,4) L-3,3 L-2,3(-2,3) L-2,2 L-1,2 (-2,1) L-1,1 Raman shift (cm

• 1)

Magnetic Field (T) L0,1 (-1,0) vF~1.025x106 m/s

E B

+

= 4 n = n

−

=1 n

+

= 3 n

+

= 2 n

+

=1 n

−

= 2 n

−

= 3 n

−

= 4 n

5 10 15 20 25 30 1000 1500 2000 2500 3000 3500 4000 4500 5000

Raman shift (cm-1) B (T)

L -1,1 L -2,1 (-1,2) L -2,2 L -3,2 (-2,3) L -3,3 "2D " band E2g phonon L 0,1 (-1,0) Beyond the standard magneto-E2g phonon resonances

Eph = E0,2

n=0

• 1
• 2

1 2

Eph = E0,1 Eph = E0,1

New class of magneto-phonon resonances:

• accelerated relaxation, shortening of the final/initial states

Interactions

E-1, 2 = E-1, 0 + Eph Ek=0 = EK, K' + EK Ek=0 = EK, K + EΓ E-1,1 = E-1, 0 + Eph = 2 x Eph

both K- and Γ-phonons invloved two-particle excitations, triple resonances intra and inter-valley scattering

• learning more on carrier dynamics

D.M. Basko, P. Leszczynski, C. Faugeras, et al

Electron – electron interactions ?

Electron – electron interactions ? Electronic inter Landau level excitations Magneto Raman scattering suspended graphene ε = 1 graphene encapsulated in hBN ε = 5 graphene on graphite ε = 10 ? EC /Ekin = ~ 2/ε PRL 114, 126804, (2015)

Electron – electron interactions ?

B n

l v n n B e v / 2 2 2 2    = = ω

Electron – electron interactions ! ε = 1 ε = 5 ε = 10 ? First order perturbation theory with respect to ε* = 3.9 ε* = 7 ε* = 12 B = 0

Electron – electron interactions ! ε = 1 ε = 5 ε = 10 ? First order perturbation theory with respect to ε* = 3.9 ε* = 7 ε* = 12 = ε + 3 Beyond FOPT

1/(N=4) expansion : RPA :

*

ε ≈

B = 0

First order perturbation theory with respect to B > 0 Denis Basko C1 = - 0.4, C2 = - 0.2 Phenomenology to match the data (numbers) ε* = 3.9 ε* = 7 ε* = 12 εδv = 1.3 εδv = 3.7 εδv = 12

N / 1

ε ≈ ε ≈

ε = 1 ε = 5 ε = 10 ?

Conclusions Magneto-optics is a useful tool to study the "unconventional" and conventional graphene structures band structure scattering efficiency electron-phonon interaction electron-electron interactions MOMB