Supporting Information
Nonlocal Electron-Phonon Coupling in the Pentacene Crystal:
Beyond the Γ-point approximation
Yuanping Yi, Veaceslav Coropceanu,* and Jean-Luc Brédas *,#
School of Chemistry and Biochemistry and Center for Organic Photonics and Electronics
Georgia Institute of Technology
Atlanta, Georgia 30332-0400
e-mail: coropceanu@gatech.edu; jean-luc.bredas@chemistry.gatech.edu
#
Also affiliated with Department of Chemistry, King Abdulaziz University, Jeddah 21589,
Saudi Arabia
Scheme S1. Illustration of the folding relation of the phonon dispersion curves of
various sizes of supercells.
Scheme S2. Frequencies of the phonon modes in the 0-1000 cm-1 energy range at the Г,
(π/a, 0, 0), and (π/2a, 0, 0) points obtained for pentacene from normal-mode calculations.
Tables S1 and S2: Estimates of G and L from the low-frequency (0-200 cm-1) and higherfrequency normal modes of the unit cell and all the supercells.
Figures S1-S5: Frequencies of the optical normal modes in the 0-200 cm-1 energy range,
spectra of phonon density of states, spectra of G2 and L, and temperature dependence of
the variances of the transfer integrals based on the unit cell and all the supercells.
S1
Phonon Energy (arb. unit)
optical
acoustic
0
a
a
a
Wave Vector
Scheme S1. Illustration of the zone-folding procedure for the phonon dispersion curves.
For instances, the phonon dispersion curves of the supercell with two unit cells are
obtained by folding the dispersion curves of the unit cell at the wavevector π/2a, and the
phonon dispersion curves of the supercell with four unit cells are obtained by folding the
dispersion curves of the supercell with two unit cells at the wavevector π/4a. Finally, all
the phonons at the mesh k-points of the unit cell can be seen at the Г-point of a supercell.
S2
400
150
350
100
300
50
250
-1
Frequency (cm )
200
0
a
a
200
600
1000
550
900
500
800
450
700
0
a
a
0
a
a
-1
Frequency (cm )
0
400
0
a
a
Wave Vector
600
Wave Vector
Scheme S2. Frequencies of the phonon modes in the 0-1000 cm-1 energy range at the Г,
(π/a, 0, 0), and (π/2a, 0, 0) points obtained for pentacene from normal-mode calculations
using the elementary unit cell and 2×1×1 and 4×1×1 supercells.
S3
Table S1. Estimates of G and L for holes and electrons (all in meV) based on the lowfrequency (0-200 cm-1) normal modes obtained from normal-mode calculations based on
the elementary unit cell and various supercells.
holes
electrons
pair 1a
pair 1b
pair 2a
pair 2b
1×1
1×2
2×1
2×2
2×4
4×2
4×4
4×8
8×4
3.2
3.5
5.0
5.2
5.2
5.3
5.3
5.3
5.3
3.3
3.6
4.9
5.3
5.2
5.3
5.3
5.3
5.3
8.2
8.3
8.6
8.9
8.9
8.9
8.9
8.9
8.9
10.8
11.5
11.6
12.2
12.2
12.3
12.4
12.4
12.4
1×1
1×2
2×1
2×2
2×4
4×2
4×4
4×8
8×4
1.3
2.0
3.7
4.4
4.5
4.8
5.1
5.1
5.1
1.4
2.1
3.0
5.0
4.6
4.8
5.1
4.9
5.0
7.6
7.4
8.2
10.9
10.1
10.3
10.4
10.3
10.3
7.6
10.9
10.6
14.2
14.5
15.1
15.9
15.9
16.1
pair 1a
pair 1b
pair 2a
pair 2b
2.7
3.0
4.4
4.5
4.5
4.6
4.7
4.7
4.7
2.8
3.0
4.3
4.5
4.5
4.6
4.7
4.6
4.7
6.3
6.7
6.5
6.7
6.8
6.8
6.8
6.8
6.8
6.9
7.5
7.3
7.6
7.7
7.7
7.7
7.7
7.8
0.9
1.4
2.8
3.1
3.1
3.4
3.7
3.7
3.7
1.0
1.5
2.3
3.3
3.2
3.4
3.6
3.6
3.6
3.4
4.3
3.7
4.6
4.7
4.6
4.9
4.9
4.9
3.5
5.0
4.4
5.7
5.9
6.0
6.4
6.4
6.5
G
L
S4
Table S2. Estimates of G and L for holes and electrons (all in meV) based on the highfrequency (>200 cm-1) normal modes obtained from normal-mode calculations based on
the elementary unit cell and various supercells.
holes
pair 1a
pair 1b
electrons
pair 2a
pair 2b
pair 1a
pair 1b
pair 2a
pair 2b
3.1
3.1
3.4
3.4
3.4
3.4
3.4
3.4
3.4
3.3
3.3
3.5
3.5
3.5
3.5
3.5
3.5
3.5
5.6
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
G
1×1
1×2
2×1
2×2
2×4
4×2
4×4
4×8
8×4
3.6
3.6
4.6
4.6
4.6
4.6
4.6
4.6
4.6
3.8
3.8
4.8
4.8
4.8
4.8
4.8
4.8
4.8
8.5
8.5
8.4
8.4
8.4
8.4
8.4
8.4
8.4
7.3
7.3
7.2
7.2
7.2
7.2
7.2
7.2
7.2
L
1×1
1×2
2×1
2×2
2×4
4×2
4×4
4×8
8×4
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.7
0.7
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.8
0.8
0.7
0.7
0.7
0.7
0.7
0.7
0.7
S5
(a) 11
(b) 12
(c) 21
(d) 22
(e) 24
(f) 42
(g) 44
(h) 48
(i) 84
0
50
100
150
-1
Frequency (cm )
200
Figure S1. Frequencies of the optical normal modes in the 0-200 cm-1 energy range
based on various supercells.
S6
0.4
0.1
0.0
-1
PhDOS (states/[cm •unit cell])
1×1
1×2
2×1
2×2
2×4
4×2
4×4
4×8
8×4
0.2
0
50 100 150 200
0.2
0.0
0
500
1000
1500
2000
-1
Frequency (cm )
Figure S2. Phonon density of states from normal-mode calculations based on various
supercells.
S7
Holes
2
G2b()
2
G2a ()
2
G1b ()
2
G1a ()
0.6
1×1
1×2
2×1
0.4
Electrons
2×2
2×4
4×2
0.6
4×4
4×8
8×4
0.4
0.2
0.2
0.0
0.0
0.4
0.4
0.2
0.2
0.0
0.0
2
1.0
1
0.5
0
0.0
4
1.0
2
0.5
0
0.0
0
500
1000 1500 2000
-1
0
500
1000 1500 2000
-1
Frequency (cm )
Frequency (cm )
Figure S3. G2() (in arbitrary units) from normal-mode calculations based on various
supercells.
S8
Holes
L2b ()
L2a ()
L1b ()
L1a ()
0.15
1×1
1×2
2×1
0.10
Electrons
2×2
2×4
4×2
0.15
4×4
4×8
8×4
0.10
0.05
0.05
0.00
0.00
0.10
0.10
0.05
0.05
0.00
0.00
0.2
0.10
0.1
0.05
0.0
0.00
0.2
0.10
0.1
0.05
0.0
0.00
0
50
100
150
200
-1
0
50
100
150
200
-1
Frequency (cm )
Frequency (cm )
Figure S4. L() (in arbitrary units) from normal-mode calculations based on various
supercells.
S9
Holes
QM: Solid lines
Cl: Dash lines
1a ()
20
1b ()
2a ()
20
10
10
1×1
1×2
2×1
0
20
2b ()
Electrons
2×2
2×4
4×2
4×4
4×8
8×4
0
20
10
10
0
30
20
10
0
30
20
10
0
0
20
10
0
20
10
0
0
100 200 300 400 500
Temperature (K)
0
100 200 300 400 500
Temperature (K)
Figure S5. Temperature dependence of the variances of the transfer integrals for holes
and electrons from normal-mode calculations based on various supercells.
S10