Supplementary material
Theoretical comparative studies on transport properties of pentacene,
pentathienoacene and 6,13-dichloropentacene
Xu Zhanga, Xiaodi Yangb,*, Hua Gengc, Guangjun Nan d, Xingwen Sun a, Xin Xua,*
a
Department of Chemistry, Fudan University, 220 Handan Road, 200433 Shanghai,
People’s Republic of China
b
Laboratory of Advanced Materials, Fudan University, 200438 Shanghai, People’s
Republic of China
c
Key Laboratory of Organic Solids, Beijing National Laboratory for Molecular
Science, Institute of Chemistry, Chinese Academy of Sciences, 100190 Beijing,
People’s Republic of China
d
Institute of Theoretical and Simulational Chemistry, Academy of Fundamental and
Interdisciplinary Sciences, Harbin Institute of Technology, 150080 Harbin, People’s
Republic of China.
1
Random walk simulation from charge transfer rates to charge mobility
Table S1 Vibrational frequencies ω (in cm-1) and relaxation energies λrel (in cm-1) of
pentacene in oxidized and reduced state.
Fig. S1 Normal modes with strong vibronic coupling in cationic and anionic pentacene.
Table S2 Vibrational frequencies ω (in cm-1) and relaxation energies λrel (in cm-1) of PTA
in oxidized and reduced state.
Fig. S2 Normal modes with strong vibronic coupling in cationic and anionic PTA.
Table S3 Vibrational frequencies ω (in cm-1) and relaxation energies λrel (in cm-1) of DCP
in oxidized and reduced state.
Fig. S3 Normal modes with strong vibronic coupling in cationic and anionic DCP.
Fig. S4 Charge hopping pathways schemes for pentacene from the crystal structure.
Fig. S5 Charge hopping pathways schemes for PTA from the crystal structure.
Fig. S6 Charge hopping pathways schemes for DCP from the crystal structure.
2
Random walk simulation from charge transfer rates to charge mobility
We perform a numerical benchmark for various approaches to simulate the charge carrier
mobility based on the charge transfer rate theory. We include the Pauli master equation
(PME) approach, kinetic Monte-Carlo (KMC) approach, the variable step size (VSS)
approach, and the first reaction method (FRM). The study is based on a one-dimensional
molecular crystal model with only the nearest intermolecular coupling and the Marcus
charge transfer rate. The following study is quite general, and can be easily extended to
study more complex systems.
PME approach:
(1) The charge carrier population at each molecular site, Pi, is initialized.
(2) One solves a set of linear equations, 𝑑𝑃𝑖 (𝑡)/𝑑𝑡 = − ∑𝑗 𝑃𝑖 𝑘𝑖𝑗 + ∑𝑗 𝑃𝑗 𝑘𝑗𝑖 , to get the
time evolution of Pi, where kij is the charge transfer rate between molecules i and j.
(3) The mobility is calculated through 𝜇 = 𝑒𝐷/𝑘𝐵 𝑇, where 𝐷 = 𝑑(∑𝑖 𝑃𝑖 𝑖 2 )/(2𝑑𝑡).
KMC approach:
(1) The charge carrier is initialized at molecule i.
(2) The next position of the charge carrier is chosen randomly from the neighbors with a
probability 𝑃𝑖𝑗 = 𝑘𝑖𝑗 / ∑𝑙 𝑘𝑖𝑙 . The time increment for such configuration change is ∆𝑡 =
1/ ∑𝑙 𝑘𝑖𝑙 , where the sum goes over all possible neighbors. One repeats this process if the
simulation stop condition is not fulfilled.
(3) One performs steps (1) and (2) for a certain number of times to get equilibrium
diffusion of the charge carrier.
(4) The mobility is calculated through 𝜇 = 𝑒𝐷/𝑘𝐵 𝑇 , where 𝐷 = 𝑑 < 𝑙 2 >/(2𝑑𝑡) ,
where < 𝑙 2 > is mean squared displacement.
VSS approach:
(1) The charge carrier is initialized at molecule i.
(2) The next position of the charge carrier is chosen randomly from the neighbors with a
probability 𝑃𝑖𝑗 = 𝑘𝑖𝑗 / ∑𝑙 𝑘𝑖𝑙 . The time increment for such configuration change is ∆𝑡 =
−ln⁡(𝑟)/ ∑𝑙 𝑘𝑖𝑙 , where the sum goes over all possible neighbors and r is a random number
between 0 and 1. One repeats this process if the simulation stop condition is not fulfilled.
3
(3) One performs steps (1) and (2) for a certain number of times to get equilibrium
diffusion of the charge carrier.
(4) The mobility is calculated through 𝜇 = 𝑒𝐷/𝑘𝐵 𝑇 , where 𝐷 = 𝑑 < 𝑙 2 >/(2𝑑𝑡) ,
where < 𝑙 2 > is mean squared displacement.
FRM approach:
(1) The charge carrier is initialized at molecule i.
(2) Different possible jumps of the charge carrier from an initial molecular i to a final
molecular j are listed with the time increment needed for the jump, ∆𝑡𝑖𝑙 = −ln⁡(𝑟)/𝑘𝑖𝑙 ,
where r is a random number between 0 and 1. From the list generated, the neighbor j with
the smallest Δtij is chosen for next position of the charge carrier. The increment time is set
as Δtij. One repeats this process if the simulation stop condition is not fulfilled.
(3) One performs steps (1) and (2) for a certain number of times to get equilibrium
diffusion of the charge carrier.
(4) The mobility is calculated through 𝜇 = 𝑒𝐷/𝑘𝐵 𝑇 , where 𝐷 = 𝑑 < 𝑙 2 >/(2𝑑𝑡) ,
where < 𝑙 2 > is mean squared displacement.
Parameters:
The nearest intermolecular distance is 5 Å, the corresponding transfer integral is 50 meV,
the temperature is set to 300 K, and five different reorganization energies, λ= 100, 200,
300, 400, and 500 meV, are chosen.
Results:
The calculated mobilities (in cm2/Vs) with different approaches and different
reorganization energies are given below. It is easy to see that all approaches give the
same results with a small numerical error. This shows that the time increment in the
KMC approach needs to sum over all possible neighbors.
λ
100
200
300
400
500
PME
14.46
1.96
0.41
0.10
0.027
KMC
14.78
1.92
0.43
0.10
0.027
VSS
14.65
1.98
0.41
0.10
0.027
FRM
14.79
1.90
0.40
0.099
0.027
4
Table S1 Vibrational frequencies ω (in cm-1) and relaxation energies λrel (in cm-1) of pentacene in
oxidized and reduced state.
Hole transfer
Neutral
ω
264
765
799
1025
1027
1189
1217
1345
1425
1448
1506
1570
1591
1593
3177
3181
3206
Electron transfer
Cation
λrel
7.3
0.8
2.1
0.3
4.7
13.0
55.3
0.7
104.7
17.8
3.6
144.2
19.4
0.1
0.1
0.5
1.3
ω
263
613
765
807
1045
1046
1200
1227
1339
1425
1430
1441
1441
1515
1560
1590
3195
3202
3224
Neutral
λrel
7.0
0.1
0.4
1.5
0.1
2.9
20.5
54.4
5.6
3.8
0.1
11.3
127.1
0.1
89.9
53.2
0.4
0.4
1.0
ω
264
616
646
765
799
1025
1027
1189
1217
1345
1425
1448
1506
1570
1591
3171
3177
3181
3206
Anion
λrel
140.5
27.0
0.3
10.8
5.8
0.2
2.6
22.7
67.0
1.5
148.5
50.8
6.3
27.6
10.1
0.1
0.4
1.6
2.8
ω
263
616
638
756
800
1041
1176
1219
1325
1409
1415
1426
1511
1548
1580
3140
3152
3179
λrel
142.6
26.1
0.8
9.4
5.9
1.0
23.5
56.4
7.0
6.0
0.1
198.9
0.5
36.4
7.6
0.1
1.8
3.3
5
1227cm-1
263cm-1
1441cm-1
1219cm-1
1560cm-1
1426cm-1
1590cm-1
Cationic state
Anionic State
Fig. S1
6
Table S2 Vibrational frequencies ω (in cm-1) and relaxation energies λrel (in cm-1) of PTA in oxidized
and reduced state.
Hole transfer
Neutral
Electron transfer
Cation
Neutral
Anion
ω
λrel
ω
λrel
ω
λrel
ω
λrel
104
235
415
438
493
642
785
902
973
1117
1205
1302
1391
1417
1439
1486
1534
3233
3269
3.5
195.8
86.8
2.8
217.6
11.6
51.4
36.2
0.2
1.8
19.9
203.8
12.9
1.1
0.1
374.8
27.6
0.4
0.4
107
239
419
436
489
640
732
781
902
964
1116
1200
1261
1351
1440
1460
1509
1535
3247
3265
3.8
202.3
84.7
17.8
202.9
13.1
0.6
45.7
39.6
1.2
1.1
83.5
154.6
41.5
0.3
6.5
0.1
331.3
0.5
0.3
235
415
438
493
642
731
785
902
973
1117
1205
1302
1417
1439
1486
1534
3233
3269
20.3
29.2
22.4
95.4
3.4
7.1
82.6
25.1
0.6
12.7
0.2
10.5
0.1
7.3
743.6
37.6
1.7
0.8
233
411
426
471
621
708
755
888
942
1092
1250
1341
1431
1474
1530
3199
3264
21.3
19.0
30.8
83.2
3.7
7.5
82.8
24.7
0.4
10.6
25.0
1.9
25.1
0.1
734.8
1.4
0.8
7
239cm-1
489cm-1
1530cm-1
1261cm-1
1535cm-1
Anionic State
Cationic state
Fig. S2
8
Table S3 Vibrational frequencies ω (in cm-1) and relaxation energies λrel (in cm-1) of DCP in oxidized
and reduced state.
Hole transfer
Neutral
Electron transfer
Cation
Neutral
Anion
ω
λrel
ω
λrel
ω
λrel
ω
λrel
264
329
686
805
912
1030
1196
1238
1344
1408
1457
1507
1554
1594
3188
3213
1.0
15.5
10.4
1.4
46.3
3.4
9.4
82.2
8.0
36.2
44.5
3.0
152.8
26.0
0.1
1.1
262
332
682
814
936
1048
1205
1247
1327
1428
1437
1510
1547
1593
3208
3230
0.9
15.1
10.2
1.0
51.6
1.8
13.6
87.9
2.5
53.5
29
28.5
116.4
41.5
0.2
0.9
264
329
623
686
805
912
1030
1196
1238
1344
1408
1457
1507
1554
3188
3213
3230
120.3
29.0
18.8
2.8
32.8
47.2
2.4
25.4
49.7
2.2
168.9
42.3
0.7
32.3
0.5
2.6
0.3
264
319
623
677
805
882
1044
1183
1240
1321
1405
1423
1510
1537
3158
3186
3212
118.4
27.4
18.7
3.3
36.5
33.7
1.0
27.1
40.1
24.9
99.1
80.6
10.6
42.7
0.4
2.9
0.4
9
936cm-1
264
1247
1405
1427
1423
1547
Anionic State
Cationic state
Fig. S3
10
Fig. S4
11
Fig. S5
12
Fig. S6
13