Transcript DFT calculations and analytical description of phonon

Interpretation of the Raman spectra of
graphene and carbon nanotubes: the effects
of Kohn anomalies and non-adiabatic effects
S. Piscanec
Cambridge University Engineering Department: Centre for Advanced Photonics and
Electronics, Cambridge, UK
G-band in graphite and nanotubes
Graphite:
one single sharp G peak
corresponding to q==0,
mode E2g
Nanotubes:
• Two main bands, G+ and G-.
• Modes derived from graphite E2g
• Metallic  semiconducting
Common interpretation: curvature
Jorio et al. PRB 65, 155412 (2002)
G+: no diameter
dependence
 LO axial
G- diameter dependence  TO
circumferential
Common interpretation: Fano resonance
In metallic tubes the G- peak is:
G
• Downshifted
+
G
• Broader
• Depends on diameter
Metallic
SWNT
Interpretation
1450 1500 1550 1600 1650 1700
-1
Raman Shift (cm )
• Fano resonance
• Phonon-Plasmon interaction
Electron-phonon coupling and Kohn
anomalies have to be considered
Kohn anomalies
• Atomic vibrations are screened by electrons
• In a metal this screening abruptly changes for vibrations
associated to certain q points of the Brillouin zone.
• Kink in the phonon dispersions: Kohn anomaly.
• Graphite is a semi-metal
• Nanotubes are folded graphite
• Nanotubes can as well be metallic
Kohn anomalies: when?
Everything depends on the geometry of the Fermi surface
Fermi surface
q = phonon wavevector
k2 = k1+ q
q
k = electron wavevector
k1
1. k1 & k2= k1+q on the Fermi surface
2.
Tangents to the Fermi surface at k1 and k2= k1+ q are parallel
•W. Kohn, Phys. Rev. Lett. 2, 393 (1959) bold
Kohn anomalies in graphite
•Graphite is a semi metal:
•Fermi surface = 2 points: K and K’ = 2 K
K’
K
p*
E



K


EF

p
Kohn Anomalies for:
•
•
q = K-K = 0 = 
q = K’-K = 2K - K = K
Kohn anomalies in graphite
IXS data: J. Maultzsch et al. Phys. Rev. Lett. 92, 075501 (2004)
-1
Frequency (cm )


K

1700
1600
A’1
1500
1400
1300
E2g
Calculations
IXS data
1200
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Phonon Wave Vector (2p/a0)
1.4
• 2 sharp kinks for modes E2g at  and A1’ at K
Kohn Anomaly
EPC ≠ 0
E2g
Kohn anomalies in nanotubes
Metallic tubes: same geometrical
conditions as graphite
p*
Ef
p
•Metallic tubes: two Giant Kohn anomalies predicted
•Semi-conducting tubes: NO Kohn anomalies predicted
1610
-1
TO:
1540
• Circumferential
• No KA
•  G+
1470
1400
1330
1260
1190
0.0
LO:
• Axial
• strong EPC
•  G-
0.1
0.2
0.3
0.4
Phonon Frequency (cm )
Metallic tubes: LO-TO splitting
0.5
Phonon Wavevector (2p/a units)
Opposite
Interpretation
10
Dynamic Effects
• Frozen phonons
• Finite differences
• Density functional perturbation theory
Rely on Born-Oppenheimer
approximation: electrons
see fixed ions
Static
approaches
For 3D crystals this is 100% OK
This is no longer true for 1D systems
• The dynamic nature of phonons can be taken into account
• Beyond Born-Oppenheimer…
Dynamic effects in nanotubes
0.00
1610
0.01
0.02
0.03
0.04
0.05
0.06
0.07
a)
1600
(11,11)
315K
1580
-1
Phonon Frequency (cm )
•KA@LO: smeared
•New KA@TO
LO
1590
1570
1560
Dynamic
Static
EZF (static)
1550
1540
1650
b)
(11,11)
315K
TO
1620
1590
•LO: increased
•TO: decreased
1560
1530
Dynamic
Static
EZF (static)
1500
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Phonon wavevector (2p/a0 units)
0.07
Dynamic effects
Phonons are not static deformations
0.00
0.02
0.03
0.04
0.05
0.06
0.07
0.00
1590
0.00
1620
0.01
0.02
0.03
0.04
0.05
a)
1590
LO
1570
1560
-1
Dynamic
Static
EZF (static)
1550
1540
b)
(11,11)
315K
TO
1620
1560
1550
-1
1570
Phonon Frequency (cm )
(11,11)
315K
1580
-1
0.03
1580
1590
Phonon Frequency (cm )
0.02
a)
a)
1600
1650
0.01
LO
T=30K
T=300K
T=1000K
1750
1700
Phonon Frequency (cm )
1610
0.01
b)
1650
1590
1530
1710
d=0.8 nm
d=1.6 nm
d=2.4 nm
b)
1620
1600
1560
1530
0.01
0.02
0.03
0.04
0.05
0.06
Phonon wavevector (2p/a0 units)
•KA@LO: smeared
•New KA@TO
1530
1550
Dynamic
Static
EZF (static)
1500
0.00
LO
1560
1500
0.07
0.00
TO
TO
1440
0.01
0.02
0.03
Phonon Wavevector (2p/a0 units)
•T increases:
•KA@LO: weaker
•KA@TO: no changes
0.00
0.01
0.02
0.03
0.04
Phonon Wavevector (2p/a0 units)
•d increases:
•KA@LO: weaker
•KA@TO: weaker
0.05
LO and TO frequencies
1620
1600
TO
1580
TO
LO
Phonon Frequency (cm-1)
1560
LO
1540
LO
Dynamic
+ curvature
Refolding-static
Dynamic
effects
Metallic
Metallic
1520
1500
1590
LO LO
1580
1570
TO
LO
TO
1560
1550
TO
1540
0.8
1.0
Dynamic + curvature
Semiconducting
Semiconducting
1.2
1.4
1.6
1.8
Diameter (nm)
2.0
2.2
2.4
Th Vs Exp: Room Temperature
1600
G+
1590
Metallic tubes
150
-1
FWHM(G ) (cm )
TO
1580
100
Raman Shift (cm-1)
-
1570
Brown [10]
Jorio [11]
Maultzsch [14]
Oron-Carl [17]
Doorn [18]
G-
1560
1550
1540
LO
1530
Metallic
0
0.6
1600
1590
1580
LO
1550
1.2
1.4
1.6
1.8
2.1
• Metallic tubes:
G-LO & G+TO
• Semiconducting tubes:
G- TO & G+ LO
Semiconducting
1.0
1.5
G-
1540
1520
0.8
1.2
Diameter (nm)
TO
1530
0.9
G+
1570
1560
50
1.8
Diameter (nm)
2.0
2.2
2.4
• Fermi golden rule:
•EPC FWHM(G-)
2.4
Interpretation of Raman spectra
TO – circumferential
LO – axial
1592
Semiconducting:
+
G
1570
• LO-TO splitting  curvature
• G+  axial
• G-  circumferential
-
G
Semiconducting SWNT
1450 1500 1550 1600 1650 1700
-1
Raman Shift (cm )
LO – axial
TO – circumferential
1550
1587
G
G
-
Metallic:
+
• LO-TO splitting  Kohn an.
• G+  circumferential
• G-  axial (KA)
• FWHM(G-)  EPC
Metallic
SWNT
1450 1500 1550 1600 1650 1700
-1
Raman Shift (cm )
Piscanec et al. PRB (2007)
G- interpretation: EPC and not
Phonon-plasmon resonance
G- band Vs T: experiments
• Metallic SWNTs
• Dielectrophoresis
• HiPCo SWNTs (Houston), d~1.1nm
• Vpp = 20 V and f=3MHz
• Raman Spectroscopy
•  = 514 nm (resonant with semicon.)
•  = 633 nm (resonant with metallic)
• Linkam stage: 80K < T < 630K
Krupke et al. Science 301, 344 (2003)
G- band Vs T: experiments
• Semiconducting tubes: G+ - G- constant  Anharmonicity
• Metallic tubes: G+ - G- increases with T  ??? (EPC)
Th Vs Exp: Temperature Dependence
70
static
dynamic
65
60
-1
45
+
G -G (cm )
50
-
55
Metallic
d=1.0nm
40
35
Semiconducting
d=1.1nm
30
25
0
150
300
450
600
750
900
Temperature (K)
Metallic tubes from R. Krupke
Conclusions
• Measurement of the Raman G-band Vs T
 Metallic tubes from dielecrophoresis
 Semiconducting tubes  G+ - G- = constant
 Metallic tubes  G+ - G- changes with T
• Kohn anomalies and electron phonon coupling
and dynamic effects
 Interpretation of G-band in SWNTs Raman spectra
 Explanation of the T-dependence of the G- in metallic
SWNTs
•