Probing Electron-Phonon Interactions
at the Saddle Point in Graphene
A. Roberts1,2, R. Binder1,3, N. H. Kwong1, D. Golla3,
D. Cormode3 , B.J. LeRoy3, H.O. Everitt2, A. Sandhu1,3
1College
of Optical Sciences, University of Arizona
2U.S. Army, Redstone Arsenal, AL
3Department of Physics, University of Arizona
Supported by DoD SMART, ARL, NSF
Γ
E(eV)
M
Conduction band
K
M-point optical
excitation
Valence band
kx
ky
Photoabsorption
M-point : Absorption peak
K point: Flat
absorption profile
∆A > 0
∆T < 0
∆A, ∆T very small
∆A < 0
∆T > 0
Photon Energy
Most optical studies
at K point (Dirac point)
This study: M point
(saddle point)
Γ
E(eV)
M
Conduction band
K
M-point optical
excitation
Pump-probe
spectroscopy at M-point:
Valence band
 Alternative to well-studied K-point
kx
k
y effects with sensitive absorption peak renormalization
 Observe many-body
Photoabsorption
 Optical observation of electron-phonon interaction
M-point : Absorption peak
Why interest in acoustic phonons?
 Measurement sensitive to electron-phonon interaction with acoustic phonons
∆A > 0
Effective acoustic deformation potential
ac
eff
K point: Flat
absorption
profile theory
~
2.8 − 7 eV
ac
eff
transport
~∆A,
16∆T− very
50 eV
small experiment, mostly electrical∆T
>0
D
D
∆T < 0
∆A < 0
ac 2
Most observables ~| Deff |
Uncertainty in observables >270
Photon
Energy transport applications
Has implications for optical and
electrical
Most optical
studies
This study:
M point
Deformation
potential
= fundamental material
parameter,
at K point
(Diracheat,
point)
(saddle point)
determines
specific
mean free path etc.
pump pulse
probe pulse
detector
Pump/probe:
~100fs duration
up to 500ps delay
degenerate at 1.6, 3.2, 4.8eV
non-degenerate at M-point
(probe scanned)
Graphene sample:
CVD-grown layers
10 individually grown layers stacked onto
sapphire substrate
Stacking does not influence monolayer behavior
(verified through Raman spectroscopy)
Saddle-point (M-point) spectroscopy
8000
occupation at T=0K
Band structure
4000
f +1 (k ) = 0
2000
0
-2000
-4000
---
-6000
-8000
K
M
-
-- - -
-
-
-
f −1 (k ) = 1
Γ
K
2000
Energy (meV)
Energy (meV)
6000
1000
0
--
-1000
-2000
K
Band structure
relevant for K and M-point
spectroscopy:
- -- - - -M
Absorption in
thermal equilibrium
-
1000
0
-1000
--
-2000
ω
- --
K
Susceptibility:
Transmission:
- - -
1000
ω
--- - - - -
0
-1000
-2000
K
M
χ (ω ) equil
M
χ (ω ) excited
T0
Differential Transmission:
-
2000
Energy (meV)
2000
Energy (meV)
Absorption in excited system
(bands shifted and more broadened)
T
T − T0
T0
Optical spectra
χ (ω )
2e 2
d 2k
f −1 (k ) − f1 (k )
2
ˆ
[
(
)
]
LF (k )
χ (ω ) =
− 2 2 2∫
〈
⋅
〉
p
k
e
θ
m ω BZ (2π ) 2
ω − ∆E (k ) + iγ (ω , k )
average over angles between flake
orientation and opt. polarization
 Transition energies:
phenomenological Fano
shape factor (exciton effect)
∆E=
(k ) E1 (k ) − E−1 (k ) + Re ∆Σ1 (k ) − Re ∆Σ −1 (k )
=
γ (ω , k )
γ 0 (ω )
− Im ∆Σ1 (k ) − Im ∆Σ −1 (k )
e-ph scattering w/o excitation
(adjusted to match exp. linear
absorption)
pump-induced changes
 Transmission calculated from optical transfer matrix approach
 General remarks on optical matrix elements and pˆ =
im
[ H 0 , r ]− see Gu et al. 2013

Fano lineshape factor LF from: Mak, Shan, Heinz, PRL 106, 046106 (2011)
Optical matrix elements: Gu, Kwong, Binder, PRB 87, 125301 (2013)
Renormalization
Σ
e− ph
s s'
Σ
s' s
e −e
Energy (meV)
2000
s=+1
1000
0
-
-1000
s=-1
-2000
s s'
e-ph
Σ e-ph = Σ e-ph
+
∆Σ
equil
s' s
K
M
changes due to modified phonon populations (temperature change)
and modified electron populations
does not require
requires thermal phonon
thermal phonon population,
population, favors
optical phonons dominate
acoustic phonons
T − T0
=
Signal
~ ∆Σ e-ph
T0
optical signal sensitive
to acoustic phonons
Renormalization due to electron-phonon interaction
Csac,s ' (=
k , k' )
2
A0
ac
D
'
u
u
|
|
|
−
〈
〉
k
k
eff
reduced to BZ
k ,s
k ', s '
4mωkLA−k'
Csop,s ' (k , k=
')
2
A0
ac
D
u
u
|
〈
〉
eff
k ,s
k ', s '
4mωkLO−k'
s s'
s' s
Emission possible without thermal excitation,
optical phonons dominate
Σ
e-ph
s
(k )


1 − f s ' (k ') + nµph (k − k' )
f s ' (k ') + nµph (k − k' )
d 2k '
µ
+
C
(
,
'
)
k
k
 (0)

s ,s '
µ
µ
(0)
(0)
(0)
∫ (2π )2 ∑
E
(
)
E
(
')
i
E
(
)
E
(
')
i
k
k
k
k
ω
η
ω
η


−
−
+
−
+
+
µ ,s '
k − k'
k − k'
s'
s
s'
 s

phonon emission
phonon absorption
Phonon emission contribution requires thermal occupation!
Favors acoustic phonons!
∆Σ e-ph
(k )
s
ph
ph


−∆
+
∆
−
∆
+
∆
f
(
k
')
n
(
k
k
'
)
f
(
k
')
n
( k − k' )
d 2k '
s
µ
s
µ
'
'
µ
+ (0)
Cs ,s ' (k , k' )  (0)

µ
µ
(0)
(0)
∫ (2π )2 ∑
E
(
k
)
E
(
k
')
i
E
(
k
)
E
(
k
')
i

ω
η

ω
η
−
−
+
−
+
+
µ ,s '
k − k'
k − k'
s'
s
s'
 s

∆f s (k=
,τ ) f s (k ,τ ) − f sequi (k ), ∆nµph=
(q ) nµph (q, T µ ) − nµph (q ,Tequi ), 〈uk ,s | u=
1/ 2 (1 + ss ' ei (φ ( k )−φ ( k' )) )
k ', s ' 〉
Saddle-point (M-point) spectroscopy: Results
∆T/T (10-2)
4.0
1.5
1.0
0.5
0.0
-0.5
τ = 4ps
600K
∆T/T (10-2)
3.5
1.5
1.0
0.5
0.0
-0.5
τ = 40ps
550K
4.5
5.0
Experiment:
Zero-crossing at ~4.1eV
Similar shape for all delay times (suggesting similar
physical origin)
Theory reproduces
 order of magnitude
 shape
∆T/T (10-2)
 zero-crossing
1.5
1.0
0.5
0.0
-0.5
τ = 400ps
350K
3.5
ac
Deff
= 5.3 eV Kaasbjerg et al., PRB 85,165440 (2012)
op
Deff
= 11 eV/A Low et al. PRB 86, 045413 (2012)
4.0
4.5
5.0
Energy (eV)
Red line: LA
Green line: LA+LO
Roberts, Binder, Kwong, Golla, Cormode, LeRoy, Everitt, Sandhu, Phys. Rev. Lett. 112, 187401 (2014)
Saddle-point (M-point) spectroscopy: Results
ac
Deff
= 20 eV:
∆T/T (10-2)
LA only
12
ac
Deff
= 20 eV
τ = 4ps
600K
8
4
0
3.5
4.0
4.5
 Lineshape wrong
(ratio positive vs negative ΔT/T)
 Zero-crossing too low
 Order of magnitude wrong
5.0
Photon Energy (eV)
∆T/T (10-2)
Electron-electron interaction (screened Hartree-Fock):
1.5
1.0
0.5
0.0
-0.5
τ = 4ps
600K
 Lineshape completely wrong
 Order of magnitude wrong
x50
3.5
4.0
4.5
Photon Energy (eV)
5.0
SHF with thermally
excited electrons
Both do not reproduce experiment
Roberts, Binder, Kwong, Golla, Cormode, LeRoy, Everitt, Sandhu, Phys. Rev. Lett. 112, 187401 (2014)
Summary
Presented differential optical transmission spectra from transitions close to M-point
Observed π-π* band renormalization
For time delays > 4 ps, electron-phonon interaction dominates
Differential transmission geometry favors acoustic phonons as source of ΔT/T
Studied electron-LA deformation potential (range of literature values ~2-50 eV)
Initial (at 4 ps) temperature determined by phonon energy and deposited optical energy
Temperature for 40ps and 400ps obtained from time-dependent data with 190 ps rate
Spectra show negative and positive signal with crossing at ~4.1eV
Good theory-experiment agreement with effective acoustic deformation potential ~5eV
Electron-electron effects do not reproduce experiment
Reference: Roberts, Binder, Kwong, Golla, Cormode, LeRoy, Everitt, Sandhu, Phys. Rev. Lett. 112, 187401 (2014)
Future improvements may include:
o
o
o
o
o
Substrate dependence of deformation potentials
Excitonic effects in probe absorption
Excitonic effects in electron-phonon coupling
More detailed electron-phonon coupling models (def. potential plus gauge field coupling)
Substrate dependence of gauge field coupling