Impact of Phonon Dispersion on Nonlocal Electron ... - ACS Publications

Subscriber access provided by READING UNIV

Article

Impact of Phonon Dispersion on Nonlocal Electron-Phonon Couplings in Organic Semiconductors: The Naphthalene Crystal as a Case Study Zeyi Tu, Yuanping Yi, Veaceslav Coropceanu, and Jean-Luc Bredas J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b08554 • Publication Date (Web): 05 Dec 2017 Downloaded from http://pubs.acs.org on December 10, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Impact of Phonon Dispersion on Nonlocal Electron-Phonon Couplings in Organic Semiconductors: The Naphthalene Crystal as a Case Study Zeyi Tu,†, ‡ Yuanping Yi,*,†, ‡ Veaceslav Coropceanu,*,§ and Jean-Luc Brédas§

†

CAS Key Laboratory of Organic Solids, CAS Research/Education Center for Excellence in Molecular Sciences, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, China ‡

University of Chinese Academy Sciences, Beijing, 100049, China

§

School of Chemistry and Biochemistry and Center for Organic Photonics and Electronics, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, United States

*

Corresponding authors. E-mail: [email protected], [email protected].

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT Recent studies point to the impact that the dispersion of both optical and acoustic phonons can have on the nonlocal electron-phonon couplings in organic molecular semiconductors. Here, in order to further elucidate the influence of phonon dispersion, we have calculated the phonon modes in the entire Brillouin zone of the naphthalene crystal. The results demonstrate that the overall nonlocal couplings are underestimated by calculations in which only the phonon modes derived at the Brillouin zone center are considered. Moreover, the contributions of acoustic phonons to the overall strength of nonlocal electron-phonon couplings are calculated to be quantitatively very significant for parallel-stacked dimers, as high as 40% for holes and 47% for electrons.

2


Page 2 of 22

Page 3 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


I. INTRODUCTION Although a comprehensive understanding of the charge-transport mechanisms in organic semiconductors has not been reached yet, there is general consensus that the subtle interplay between electronic couplings and electron-phonon couplings plays a crucial role.1-8 In the framework of a simple tight-binding description, two major electron-phonon coupling mechanisms are distinguished:1 the local or Holstein-type coupling9,10 and nonlocal or Peierls-type coupling.11 The local coupling is due to the modulation of the site energy by vibrations; its strength is expressed by the polaron binding energy Epol or, in the context of Marcus electron-transfer theory,12 by the reorganization energy λ (≈ 2Epol). The nonlocal coupling arises from the dependence of the transfer integral (electronic coupling) tmn on the distances between adjacent molecules and their relative orientations; this represents the major interaction in Peierls-type models, such as the Su-Schrieffer-Heeger Hamiltonian.13 Recent studies have shown that the magnitudes of the nonlocal couplings strongly influence the charge carrier mobilities14-18 and their temperature dependence.19-25 In addition, the nonlocal couplings also contribute to the simultaneous presence of band-like transport behaviors and incoherent states that are dynamically localized by thermal lattice disorder.26,27 However, most estimates of the nonlocal coupling constants reported to date are based exclusively on Г-point phonon modes28-33 or a few dispersive phonon branches.34 Although the dispersion of all phonon branches were considered in the work of Stojanovic and co-workers, their approach does not allow one to differentiate between Holstein-type and Peierls-type 3



Page 4 of 22

contributions;35 that information is, however, important in the development of polaron charge-transport models. In our previous work, we have assessed the influence on the nonlocal couplings due to phonon modes beyond the Г point by using a supercell approach.36 The results suggested that the overall nonlocal couplings are substantially underestimated by calculations taking sole account of phonons at the Г point of the unit cell. Yet, a complete consideration of the phonon dispersion would require calculations performed on very large supercells, which makes this approach impractical. Here, the phonon dispersion is calculated by means of a direct method based on a small-size supercell and the impact of phonon dispersion on the nonlocal couplings in organic semiconductors is investigated by taking the naphthalene crystal as a representative example.

II. METHODOLOGY In the harmonic approximation, the Hamiltonian for the lattice vibrations and atomic displacements are given by: 1

𝐻𝐿 = 2 ∑𝒌,𝑗 𝑄̇ (𝒌, 𝑗)𝑄̇ ∗ (𝒌, 𝑗) + 𝜔2 (𝒌, 𝑗)𝑄(𝒌, 𝑗)𝑄∗ (𝒌, 𝑗) 𝒖(𝑛, 𝜇) =

1 √𝑁𝑚𝜇

∑𝒌,𝑗 𝒆(𝒌, 𝑗, 𝜇)𝑒𝑥𝑝[𝑖 ∙ 𝒌𝑹(𝑛, 𝜇)]𝑄(𝒌, 𝑗)

(1) (2)

Here, 𝒌 represents the phonon wave vector; 𝑗, the phonon branch index; 𝜔 and 𝑄, the phonon frequency and normal coordinates, respectively; 𝑁, the total number of unit cells; 𝑹(𝑛, 𝜇), the position of the 𝜇-th atom in the 𝑛-th unit cell; 𝑚, the atomic

4


Page 5 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


mass; 𝒆, the polarization vector. The complex normal coordinates 𝑄(𝒌, 𝑗) can be expressed in terms of real normal coordinates {𝑄(𝑘, 𝑗, 𝑠); 𝑠 = 1, 2}:37 𝑄(𝒌, 𝑗) = 𝑄(𝒌, 𝑗, 1) + 𝑖 ∙ 𝑄(𝒌, 𝑗, 2)

(3)

Then, we obtain: 1

𝐻𝐿 = ∑𝒌,𝑗,𝑠 𝑄̇ 2 (𝒌, 𝑗, 𝑠) + 𝜔2 (𝒌, 𝑗)𝑄 2 (𝒌, 𝑗, 𝑠)

(4)

2

By making use of dimensionless coordinates: 𝜔(𝒌,𝑗)

𝑞(𝒌, 𝑗, 𝑠) = √

ℏ

𝑄(𝒌, 𝑗, 𝑠)

(5)

the lattice Hamiltonian 𝐻𝐿 is rewritten as: 𝒬̇2 (𝒌,𝑗,𝑠)

1

𝐻𝐿 = 2 ∑𝒌,𝑗,𝑠 ℏ𝜔(𝒌, 𝑗) [ 𝜔2(𝒌,𝑗) + 𝑞 2 (𝒌, 𝑗, 𝑠)]

(6)

and within the first-order Taylor expansion, the dependence of transfer integrals on the normal coordinate can be given as (0)

𝑡𝑚𝑛 = 𝑡𝑚𝑛 + ∑𝒌,𝑗,𝑠 𝜐𝑚𝑛 (𝒌, 𝑗, 𝑠) 𝑞(𝒌, 𝑗, 𝑠)

(7)

Here, ℏ denotes Planck’s constant; 𝑡𝑚𝑛 = 〈𝛹𝑚 |𝐻|𝛹𝑛 〉 , the transfer integral involving two charge-localized states 𝛹𝑚 and 𝛹𝑛 on molecules 𝑚 and 𝑛 of the dimer system, and in the one-electron approximation, the localized electronic states and the electronic Hamiltonian of the system 𝐻 turn to be the frontier orbitals of (0)

isolated molecules and the Fock operator, respectively;38 𝑡𝑚𝑛 , the transfer integral computed at the equilibrium crystal geometry; and 𝜐𝑚𝑛 (𝒌, 𝑗, 𝑠) = 𝜕𝑡𝑚𝑛 ⁄𝜕𝑞(𝒌, 𝑗, 𝑠), the linear nonlocal electron-phonon coupling constant. The overall strength of the nonlocal electron-phonon coupling can be quantified 5



Page 6 of 22

by means of parameters L and G:28,29 𝐿𝑚𝑛 = ∑𝒌,𝑗,𝑠 2 𝐺𝑚𝑛 = ∑𝒌,𝑗,𝑠

2 (𝒌,𝑗,𝑠) 𝜐𝑚𝑛

(8)

2ℏ𝜔(𝒌,𝑗) 2 (𝒌,𝑗,𝑠) 𝜐𝑚𝑛

(9)

2

G and L are directly related to the variance of the transfer integrals due to thermal fluctuations:28,29 𝜎 2 = 〈(𝑡𝑚𝑛 − 〈𝑡𝑚𝑛 〉)2 〉 = ∑𝒌,𝑗,𝑠

2 (𝒌,𝑗,𝑠) 𝜐𝑚𝑛

2

ℏ𝜔(𝒌,𝑗)

𝑐𝑜𝑡ℎ (

2𝑘𝐵 𝑇

)

(10)

where 〈… 〉 represents the thermal average over the vibration manifolds, 𝑘𝐵 denotes the Boltzmann constant, and T is the temperature. In the limiting cases of low temperatures (ℏ𝜔 ≫ 𝑘𝐵 𝑇) and high temperatures (ℏ𝜔 ≪ 𝑘𝐵 𝑇), 𝐺 2; 𝜎2 = { 2𝐿𝑘𝐵 𝑇;

ℏ𝜔 ≫ 𝑘𝐵 𝑇 ℏ𝜔 ≪ 𝑘𝐵 𝑇

(11)

The geometric structure and electronic properties of the naphthalene crystal were computed

at

the

density

functional

theory

(DFT)

level

using

the

Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional,39 a plane-wave basis set (with 600 eV cutoff) and projector augmented wave potentials.40,41 All the DFT calculations were carried out using the VASP code.42 In the geometry optimization, the cell parameters were constrained to the experimental values.43 The first Brillouin zone was sampled using a 4×4×4 Monkhorst-Pack mesh in the electronic momentum space, and the Brillouin zone integration was performed with a modified tetrahedron method.44 Convergence criteria were set to 10-7 eV for electronic self-consistency cycles and 0.001 eV/Å for ionic update cycles.

6


Page 7 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The

phonon

dispersion

calculations

were

performed

by

using

the

Parlinski-Li-Kawazoe methodology implemented in the PHONOPY code,45-47 based on the Hellmann-Feynman forces calculated with the VASP code. The force constants are determined from the Hellmann-Feynman forces induced by the displacement of an atom in a sufficient large supercell.46,47 The phonon frequencies and polarization vectors are then obtained by diagonalizing the dynamical matrices.46,47 In this work, the atomic displacements are set to 0.01 Å. The regular unit cell (1×1×1 cell) as well as 2×1×1, 1×2×1, 1×1×2 and 1×2×2 supercells are used to calculate the Hellmann-Feynman forces. Correspondingly, the Brillouin zones of the above supercells are sampled by 4×4×4, 2×4×4, 4×2×4, 4×4×2 and 4×2×2 meshes for the electron momentum. The nonlocal electron-phonon coupling constants were computed by means of numerical derivations of the transfer integrals with respect to the normal coordinates. The crystal distortion steps along the normal modes were set to make the largest atomic displacement equal to 0.01 Å. Because of the large amount of necessary calculations, the transfer integrals for the dimers extracted from optimized and distorted crystal geometries were evaluated at the INDO level of theory48 with the Mataga-Nishimoto potential to describe the Coulomb repulsion term.49,50 We have confirmed that INDO provides results similar to those from PBE and B3LYP DFT approaches at the equilibrium crystal geometry (see Table S2). Nonetheless, it should be noticed that calculations based on the INDO and DFT approaches might result in different values of the nonlocal electron-phonon coupling constants. 7



III. RESULTS AND DISCUSSION The naphthalene crystal structure (see Figure 1) belongs to the monoclinic 𝑃21 ⁄𝑎 space group with two molecules per unit cell.43 As can be seen in Figures S1 and 2, the phonon dispersion curves are very dependent on the supercell sizes and expanding directions (see Supporting Information for detailed discussion). Because of the periodic boundary conditions, forces are induced by the displacement of an atom within the supercell as well as by those of the equivalent atoms in all images of the supercell. To prevent such an artificial symmetry effect, the size of supercell has to exceed the interaction range.46,47 For the studied naphthalene crystal, the “exact” phonon dispersion profiles without imaginary frequencies are then obtained by using the 1×2×2 supercell (Figure 2).

Figure 1. Naphthalene crystal structure (a=8.098Å, b=5.953Å, c=8.652Å, and β=124.4º) and illustration of the molecular pairs used in the calculations.

8


Page 8 of 22

Page 9 of 22

200

-1

Frequency (cm )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


100

0

B



Y



Z





A

D



C



E

Wavevector

Figure 2. Phonon dispersions calculated by using the 1×2×2 supercell. The k-points of high symmetry in the first Brillouin zone are labeled as follows: B = (0.5, 0, 0), Γ = (0, 0, 0), Y = (0, 0.5, 0), Z = (0, 0, 0.5), A = (0.5, 0.5, 0), D = (0.5, 0, 0.5), C = (0, 0.5, 0.5), E = (0.5, 0.5, 0.5).

There are 3 acoustic and 105 optical phonon branches for the naphthalene crystal. To verify the reliability of the direct method for phonon frequencies, Table 1 collects the PHONOPY and VASP estimates of the nine lowest-energy phonon modes at the Γ-point in comparison with the experimental values (a complete listing of the PHONOPY estimates at the Γ-point is given in Table S1; the indexes follow the order of frequencies at the Γ-point). Overall, the PHONOPY estimates are in good agreement with both the VASP estimates and experimental data.51 Our calculations point to some modest mixing between intermolecular and intramolecular modes, as indicated already by previous MM and DFT calculations.28

9



Page 10 of 22

Table 1. PHONOPY and VASP estimates of frequencies (in cm-1) of the nine lowest-energy vibration modes derived at the Γ-point. Type translation libration libration translation libration libration translation libration libration

PHONOPY 55 50 59 74 82 84 109 113 134

VASP 52 52 61 75 88 93 114 119 141

experiment51 44 56 67 75 83 88 106 121 141

The dispersions (differences between the smallest and largest values; see Figures 2 and S3 for more detailed results) of the three acoustic modes are similar, about 55-58 cm-1; these values are much larger than those of the optical modes (in the range of 0-36cm-1) and tend to decrease with increasing frequency. The phonon density of states (PhDOS) obtained from 1×1×1 (only Γ-point), 2×2×2, 4×4×4, 8×8×8 and 16×16×16 k-space grid (all include Γ-point and boundary points) are shown in Figure 3. The PhDOS in the low-frequency range (below 150 cm-1) is obviously modified when enlarging the k-space grid. As a result especially of the strong dispersions of the long-wavelength acoustic phonons, the change in PhDOS is evaluated to be very large when the frequency is below 50 cm-1. In contrast, the PhDOS in the high-frequency range (above 220 cm-1) is hardly affected by the modifications in k-space grid. By and large, the PhDOS converges when the k-space grid reaches 8×8×8.

10


Page 11 of 22

600

300

111 222 444 888 161616

200 100

400

PhDOS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 0

100

200

200

0 0

500

1000

1500

-1

Frequency (cm )

Figure 3. Phonon density of states obtained by using 1×1×1, 2×2×2, 4×4×4, 8×8×8 and 16×16×16 k-point meshes; a Gaussian broadening with a full width at half maximum (FWHM) of 10 cm-1 has been used.

The relaxation energies 𝐿 and 𝐺 for the molecular dimers along with the related transfer integral 𝑡 (0) and the ratio of 𝐿 vs |𝑡 (0) | are listed in Table 2. With increasing k-space grid, the 𝐿 values gradually converge, and the use of an 8×8×8 k-space grid is sufficient to obtain well-converged values. Clearly, in the naphthalene crystal, the values of 𝐿 are underestimated if only Γ-point normal modes are taken into consideration, especially for holes. The 𝐿 values for holes based on the Γ-point normal modes account only for 57%, 77%, and 75% of the converged values for pairs 1, 2, and 3, respectively; the corresponding values for electrons are 93%, 77%, and 100%. However, the 𝐺 values show much less dependence on the used k-space grids (see Table S2); also, they are quite large, at least one-third of the transfer integral variances (σ) at 300 K. 11



Page 12 of 22

Table 2. Estimates of 𝑡 (0) , 𝐺, and 𝐿 for holes and electrons (in meV) for various k-point meshes of phonons. Holes Pair 1

Pair 2

Electrons Pair 1

Pair 3 t

-35.9

-4.0

Pair 2

Pair 3

28.8

-1.4

9.9 14.0 11.2 12.2 12.2 13.0 12.4 12.8 12.8 12.9 12.8 12.9 12.9

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

25.8

5.3

12.0

3.4

0.45

0.36

(0)

12.4

9.3 L

1×1×1 1×1×2 1×2×1 2×1×1 2×2×2 2×2×4 2×4×2 4×2×2 4×4×4 4×4×8 4×8×4 8×4×4 8×8×8

13.2 14.8 19.1 13.7 20.5 20.4 22.6 20.6 22.8 22.8 23.0 22.8 23.1

28.2 35.5 32.3 37.1 35.7 36.2 36.2 38.0 36.6 36.7 36.6 36.9 36.7

8×8×8

34.5

43.6

8×8×8

12.7

27.3

8×8×8

0.64

9.18

1.5 1.6 1.8 2.3 2.0 2.0 2.0 2.1 2.0 2.0 2.0 2.0 2.0

12.8 14.5 12.5 13.3 13.3 13.2 13.8 13.4 13.7 13.7 13.7 13.7 13.7 σ (300 K) 10.2 26.6 G 6.1 8.4 (0) L/|𝑡 | 0.16 1.47

The dependence of 𝐿 on phonon frequencies is shown in Figure 4. The calculations based on the Г-point phonon modes underestimate the contributions of phonon modes with frequencies under 60 cm-1. For pair 1 (i.e., parallel-stacked dimer), the main contribution to the nonlocal coupling is coming from the phonon modes in the 0-100 cm-1 range. For pairs 2 and 3, phonons above 200 cm-1 also contribute to the overall nonlocal coupling, especially for holes. In previous estimates based on the Γ-point normal modes, the two phonons at around 1620 cm-1 yield very large coupling constants;28 the present calculations also show that these two phonons make an 12


Page 13 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


appreciable contribution to 𝐿 for holes in pairs 2 and 3 (see Figure S6), and a very significant contribution to 𝐺 2 (see Figure S7). Overall, however, the main contribution to 𝐿 come from low-frequency acoustic and optical phonons (the first 16 phonon branches, see Figure S4). This is in agreement with previous findings indicating that nonlocal electron-phonon interactions are dominated by intermolecular vibrations.17,21,28,29,52 The decomposition of 𝐿 and 𝐺 into the contributions from acoustic and optical phonons is shown in Table 3. For holes, the contributions of acoustic phonons to 𝐿 are as high as 40%, 37%, and 43% for pairs 1, 2 and 3, respectively; in the case of electrons, the corresponding contributions are 47%, 27%, and 20%, respectively. On the other hand, the contributions to 𝐺 2 from acoustic phonons are much smaller, about 3%-22%. Overall, these results mean that the effect of the acoustic phonons on the thermally induced fluctuations of the transfer integrals (and consequently on charge transport) increases with an increase in temperature.

13



LPair 1

Holes

Electrons 888 111

0.5

LPair 2

0.0 1.0 0.5 0.0

LPair 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 22

0.5 10

10

0.0 0

100

200

300

400

500 0

100

-1

200

300

400

500

-1

Frequency (cm )

Frequency (cm )

Figure 4. Dependence of 𝐿 on phonon frequencies. The plots are obtained by 𝜐2 (𝒌,𝑗,𝑠)

replacing the delta functions in the expression 𝐿 = ∑𝒌,𝑗,𝑠 2ℏ𝜔(𝒌,𝑗) 𝛿[𝜔 − 𝜔(𝒌, 𝑗)] by Gaussian functions with a full width at half-maximum (FWHM) of 10 cm-1.

Table 3. Contributions of acoustic and optical phonons to 𝐿 and 𝐺 for holes and electrons (in meV). Holes Pair 1

Pair 2

Pair 3

Electrons Pair 1

Pair 2

Pair 3

6.4 7.3

3.5 9.3

0.1 0.4

3.9 7.4

2.9 11.7

0.5 3.4

L acoustic optical

9.2 13.9

13.5 23.3

0.9 1.2 G

acoustic optical

4.7 11.8

5.6 26.7

1.4 5.9

IV. CONCLUSIONS We have studied the impact of phonon dispersion on nonlocal electron-phonon couplings in the naphthalene crystal. The full dispersions of both optical and acoustic phonons were estimated via first-principles calculations. We show that in order to 14


Page 15 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


describe the force constants correctly and eliminate the imaginary phonon modes, supercells of appropriately large sizes must be used for the calculations of the phonon properties. The calculations show that, overall, the contributions of acoustic phonons and optical phonons to the relaxation energies (𝐿) describing the nonlocal electron-phonon interactions are comparable for both holes and electrons. This result underlines that the interactions with both acoustic and optical phonons should be included in the polaron models in order to reach an adequate description of charge transport in organic semiconductors. In particular, the consideration of the significant couplings with the low-frequency (especially long-wavelength acoustic) phonons will be important in the understanding of the temperature dependence of charge carrier mobility. Finally, we underline that, along some directions in the naphthalene crystal, the 𝐿⁄|𝑡 (0) | values are estimated to be larger than unity, see Table 2. This result implies that the electron-phonon coupling should not be treated in the perturbation limit, at least when modeling charge transport along these specific crystal directions.

ASSOCIATED CONTENT

Supporting information Phonon dispersions calculated by using the normal unit cell, and 2×1×1, 1×2×1, and 1×1×2 supercells, illustration of the closest inter-atomic distances between adjacent equivalent molecules along the a-, b- and c-axes, calculated phonon frequencies at the 15



Γ-point, dispersions of phonons, estimates of transfer integrals in the equilibrium geometry, dependence of 𝐺 2 on different k-point meshes, dispersions and decomposition of 𝐿 and 𝐺 2 into each phonon branch and dependence of 𝐿 and G2 on phonon frequencies.

AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] (Y.Y.), [email protected] (V.C.) Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS The work in Beijing was supported by the National Natural Science Foundation of China (Grant No 21373229), the Ministry of Science and Technology of China (Grant Nos 2017YFA0204502, 2014CB643506), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No XDB12020200). The work at the Georgia Institute of Technology was partly supported by the Office of Naval Research (Grant No N00014-17-1-2008).

16


Page 16 of 22

Page 17 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


REFERENCES (1) Coropceanu, V.; Cornil, J.; da Silva Filho, D. A.; Olivier, Y.; Silbey, R.; Bredas, J. L. Charge Transport in Organic Semiconductors. Chem. Rev. 2007, 107, 926-952. (2) Ortmann, F.; Bechstedt, F.; Hannewald, K. Charge Transport in Organic Crystals: Theory and Modelling. Phys. Status Solidi B 2011, 248, 511-525. (3) Troisi, A. Charge Transport in High Mobility Molecular Semiconductors: Classical Models and New Theories. Chem. Soc. Rev. 2011, 40, 2347-2358. (4) Bässler, H.; Köhler, A.: Charge Transport in Organic Semiconductors. In Unimolecular and Supramolecular Electronics I; Metzger, R. M., Ed.; Springer Berlin Heidelberg, 2012; Vol. 312; pp 1-65. (5) Coropceanu, V.; Li, Y.; Yi, Y.; Zhu, L.; Brédas, J.-L. Intrinsic Charge Transport in Single Crystals of Organic Molecular Semiconductors: A Theoretical Perspective. MRS Bull. 2013, 38, 57-64. (6) Shuai, Z.; Geng, H.; Xu, W.; Liao, Y.; Andre, J. M. From Charge Transport Parameters to Charge Mobility in Organic Semiconductors through Multiscale Simulation. Chem. Soc. Rev. 2014, 43, 2662-2679. (7) Bakulin, A. A.; Lovrincic, R.; Yu, X.; Selig, O.; Bakker, H. J.; Rezus, Y. L.; Nayak, P. K.; Fonari, A.; Coropceanu, V.; Bredas, J. L.; Cahen, D. Mode-Selective Vibrational Modulation of Charge Transport in Organic Electronic Devices. Nat. Commun. 2015, 6, 7880. (8) Oberhofer, H.; Reuter, K.; Blumberger, J. Charge Transport in Molecular Materials: An Assessment of Computational Methods. Chem. Rev. 2017, 117, 10319-10357. (9) Holstein, T. Studies of Polaron Motion. Ann. Phys. 1959, 8, 325-342. (10) Holstein, T. Studies of Polaron Motion: Part Ii. The “Small” Polaron. Ann. Phys. 1959, 8, 343-389. (11) Peierls, R. E.: Quantum Theory of Solids; Oxford University Press: New York, 1955. 17



Page 18 of 22

(12) Marcus, R. A. Electron Transfer Reactions in Chemistry. Theory and Experiment. Rev. Mod. Phys. 1993, 65, 599-610. (13) Su, W. P.; Schrieffer, J. R.; Heeger, A. J. Solitons in Polyacetylene. Phys. Rev. Lett. 1979, 42, 1698-1701. (14) Wang, L.; Li, Q.; Shuai, Z.; Chen, L.; Shi, Q. Multiscale Study of Charge Mobility of Organic Semiconductor with Dynamic Disorders. Phys. Chem. Chem. Phys. 2010, 12, 3309-3314. (15) Vehoff, T.; Baumeier, B. r.; Troisi, A.; Andrienko, D. Charge Transport in Organic Crystals: Role of Disorder and Topological Connectivity. J. Am. Chem. Soc. 2010, 132, 11702-11708. (16) Han, G.; Shen, X.; Duan, R.; Geng, H.; Yi, Y. Revealing the Influence of the Solvent Evaporation Rate and Thermal Annealing on the Molecular Packing and Charge Transport of Dpp(Tbfu)2. J. Mater. Chem. C 2016, 4, 4654-4661. (17)Chernyshov, I. Y.; Vener, M. V.; Feldman, E. V.; Paraschuk, D. Y.; Sosorev, A. Y. Inhibiting Low-Frequency Vibrations Explains Exceptionally High Electron Mobility in 2,5-Difluoro-7,7,8,8-Tetracyanoquinodimethane (F2-Tcnq) Single Crystals. J. Phys. Chem. Lett. 2017, 8, 2875-2880. (18) Zhu, L.; Geng, H.; Yi, Y.; Wei, Z. Charge Transport in Organic Donor-Acceptor Mixed-Stack Crystals: The Role of Nonlocal Electron-Phonon Couplings. Phys. Chem. Chem. Phys. 2017, 19, 4418-4425. (19) Hannewald, K.; Bobbert, P. A. Anisotropy Effects in Phonon-Assisted Charge-Carrier Transport in Organic Molecular Crystals. Phys. Rev. B 2004, 69, 075212. (20) Hannewald, K.; Stojanović, V. M.; Schellekens, J. M. T.; Bobbert, P. A.; Kresse, G.; Hafner, J. Theory of Polaron Bandwidth Narrowing in Organic Molecular Crystals. Phys. Rev. B 2004, 69, 075211. (21) Sun, Y.; Tan, L.; Jiang, S.; Qian, H.; Wang, Z.; Yan, D.; Di, C.; Wang, Y.; Wu, W.; Yu, G.; Yan, S.; Wang, C.; Hu, W.; Liu, Y.; Zhu, D. High-Performance Transistor Based

on

Individual

Single-Crystalline

Micrometer

Perylo[1,12-B,C,D]Thiophene. J. Am. Chem. Soc. 2007, 129, 1882-1883. 18


Wire

of

Page 19 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(22) Wang, L. J.; Li, Q. K.; Shuai, Z. Effects of Pressure and Temperature on the Carrier Transports in Organic Crystal: A First-Principles Study. J. Chem. Phys. 2008, 128, 194706. (23) Ortmann, F.; Roche, S. Polaron Transport in Organic Crystals: Temperature Tuning of Disorder Effects. Phys. Rev. B 2011, 84, 180302. (24) Liu, F.; Proynov, E.; Yu, J. G.; Furlani, T. R.; Kong, J. Comparison of the Performance of Exact-Exchange-Based Density Functional Methods. J. Chem. Phys. 2012, 137, 114104. (25) Wang, L.; Prezhdo, O. V.; Beljonne, D. Mixed Quantum-Classical Dynamics for Charge Transport in Organics. Phys. Chem. Chem. Phys. 2015, 17, 12395-12406. (26) Fratini, S.; Ciuchi, S. Bandlike Motion and Mobility Saturation in Organic Molecular Semiconductors. Phys. Rev. Lett. 2009, 103, 266601. (27) Ciuchi, S.; Fratini, S. Band Dispersion and Electronic Lifetimes in Crystalline Organic Semiconductors. Phys. Rev. Lett. 2011, 106, 166403. (28) Coropceanu, V.; Sánchez-Carrera, R. S.; Paramonov, P.; Day, G. M.; Brédas, J.-L. Interaction of Charge Carriers with Lattice Vibrations in Organic Molecular Semiconductors: Naphthalene as a Case Study. J. Phys. Chem. C 2009, 113, 4679-4686. (29) Sanchez-Carrera, R. S.; Paramonov, P.; Day, G. M.; Coropceanu, V.; Bredas, J. L. Interaction of Charge Carriers with Lattice Vibrations in Oligoacene Crystals from Naphthalene to Pentacene. J. Am. Chem. Soc. 2010, 132, 14437-14446. (30)Girlando, A.; Grisanti, L.; Masino, M.; Bilotti, I.; Brillante, A.; Della Valle, R. G.; Venuti, E. Peierls and Holstein Carrier-Phonon Coupling in Crystalline Rubrene. Phys. Rev. B 2010, 82, 035208. (31) Girlando, A.; Grisanti, L.; Masino, M.; Brillante, A.; Della Valle, R. G.; Venuti, E. Interaction of Charge Carriers with Lattice and Molecular Phonons in Crystalline Pentacene. J. Chem. Phys. 2011, 135, 084701. (32) Girlando, A.; Masino, M.; Bilotti, I.; Brillante, A.; Della Valle, R. G.; Venuti, E. Phonon Dynamics and Electron-Phonon Coupling in Pristine Picene. Phys. Chem. Chem. Phys. 2012, 14, 1694-1699. 19



Page 20 of 22

(33) Ordejón, P.; Boskovic, D.; Panhans, M.; Ortmann, F. Ab Initio Study of Electron-Phonon Coupling in Rubrene. Phys. Rev. B 2017, 96, 035202. (34) Sinova, J.; Schliemann, J.; Nunez, A. S.; MacDonald, A. H. 2d Bands and Electron-Phonon Interactions in Polyacene Plastic Transistors. Phys. Rev. Lett. 2001, 87, 226802. (35) Vukmirovic, N.; Bruder, C.; Stojanovic, V. M. Electron-Phonon Coupling in Crystalline Organic Semiconductors: Microscopic Evidence for Nonpolaronic Charge Carriers. Phys. Rev. Lett. 2012, 109, 126407. (36) Yi, Y.; Coropceanu, V.; Brédas, J. L. Nonlocal Electron-Phonon Coupling in the Pentacene Crystal: Beyond the Gamma-Point Approximation. J. Chem. Phys. 2012, 137, 164303. (37) Brüesch, P.: Phonons: Theory and Experiments I; Springer-Verlag Berlin Heidelberg: New York, 1982. (38) Conwell, E. M. Band Transport in Quasi-One-Dimensional Conductors in the Phonon-Scattering

Regime

and

Application

to

Tetrathiofulvalene-Tetracyanoquinodimethane. Phys. Rev. B 1980, 22, 1761-1780. (39) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865-3868. (40) Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953-17979. (41) Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B 1999, 59, 1758-1775. (42) Kresse, G.; Furthmüller, J. Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set. Comput. Mater. Sci. 1996, 6, 15-50. (43) Ponomarev, V. I.; Filipenko, O. S.; Atovmyan, L. O. Crystal and Molecular-Structure of Naphthalene at -150 Degrees Celsius. Kristallografiya 1976, 21, 392-394. (44) Blöchl, P. E.; Jepsen, O.; Andersen, O. K. Improved Tetrahedron Method for Brillouin-Zone Integrations. Phys. Rev. B 1994, 49, 16223-16233. 20


Page 21 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(45) Togo, A.; Tanaka, I. First Principles Phonon Calculations in Materials Science. Scr. Mater. 2015, 108, 1-5. (46) Parlinski, K.; Li, Z. Q.; Kawazoe, Y. First-Principles Determination of the Soft Mode in Cubiczro2. Phys. Rev. Lett. 1997, 78, 4063-4066. (47) Togo, A.; Oba, F.; Tanaka, I. First-Principles Calculations of the Ferroelastic Transition between Rutile-Type Andcacl2-Typesio2at High Pressures. Phys. Rev. B 2008, 78, 134106. (48) Ridley, J.; Zerner, M. An Intermediate Neglect of Differential Overlap Technique for Spectroscopy: Pyrrole and the Azines. Theoretica Chimica Acta 1973, 32, 111-134. (49) Mataga, N.; Nishimoto, K. Electronic Structure and Spectra of Nitrogen Heterocycles. Zeitschrift für Physikalische Chemie 1957, 13, 140-157. (50) Nishimoto, K.; Mataga, N. Electronic Structure and Spectra of Some Nitrogen Heterocycles. Zeitschrift für Physikalische Chemie 1957, 12, 335-338. (51) Suzuki, M.; Yokoyama, T.; Ito, M. Polarized Raman Spectra of Naphthalene and Anthracene Single Crystals. Spectrochimica Acta A 1968, 24, 1091-1107. (52) Li, Y.; Coropceanu, V.; Bredas, J. L. Nonlocal Electron-Phonon Coupling in Organic Semiconductor Crystals: The Role of Acoustic Lattice Vibrations. J. Chem. Phys. 2013, 138, 204713.

21



Table of Contents

22


Page 22 of 22

Impact of Phonon Dispersion on Nonlocal Electron ... - ACS Publications

Recommend Documents