Elucidation of molecular-level charge transport in an organic amorphous system

Charge transport in organic aggregates is an inevitable process in driving charge-injection or charge-extraction type organic semiconductor devices. In organic light-emitting diodes (OLEDs), materials are used in “amorphous” states. Therefore, it is important to understand charge transports in organic “amorphous” aggregates to establish the fundamental principles and to achieve highly efficient OLEDs. There are many studies about charge transports in amorphous aggregates both theoretically and experimentally so far^{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}; however, seamless understanding of charge transport has been difficult from the molecular-level to the device-scale. Even the molecular packing structure has not been well understood due to the amorphous nature.

We have developed a multiscale charge transport simulation method by combining quantum chemical calculations, molecular dynamics (MD), and kinetic Monte Carlo (kMC) simulations^5,7,10,11. The charge hopping simulation in ref. ¹¹ successfully reproduced experimentally obtained hole and electron mobilities quantitatively without using any fitting parameters. The excellent agreement with the experiments confirms that the elementary process of charge transport in organic amorphous aggregation systems can be reasonably described by intermolecular charge hopping. In this study, we clarify the local molecular packing structures and the detailed charge hoppings among molecules in an amorphous film. To demonstrate the generality of this study, we focused on 4,4’-bis(N-carbazolyl)-1,1’-biphenyl (CBP), which has been widely used in organic devices. We find three types of charge traps, caused by (1) energy gaps between the hopping sites, (2) distribution of molecular packings, and (3) backward hopping against the electric field. We here denote them as “diagonal traps (energetic traps),” “off-diagonal traps (structural traps),” and “backward hopping traps,” respectively. This study reveals the details of charge transport from the molecular/electron level to the macroscopic device level by explicitly considering the distributed intra- and intermolecular packing structures and the distributed energy levels responsible for the distributed charge transport.

Distribution of charge mobilities

In OLEDs, analysis of the spatial distribution of charge recombination position is of crucial importance; to avoid annihilation of excitons and consequently to suppress efficiency roll-off and improve device lifetime, the charge recombination position should be spatially distributed in the emission layer. Since the charge recombination region is determined by the mobilities and trajectories of holes and electrons, the values of mobilities, especially their distributions, are of primary importance in understanding and solving the above roll-off and lifetime problems. However, until now, the charge mobility has been expressed as a constant value and mobility distribution has not been well considered.

Here, we carried out a multiscale simulation for CBP. An MD simulation was carried out to construct an amorphous aggregate, consisting of 4000 molecules in total, under the periodic boundary condition (PBC), using the density functional theory (DFT)-optimized CBP molecule as an initial structure. Hereafter, we mainly discuss hole transport; the same discussion holds also for electron transport as shown briefly in the main text and primarily in Supplementary Information. Multiple molecular orbitals (MOs) (not only highest occupied MO (HOMO) but also HOMO-1, HOMO-2, HOMO-3, …) are considered because the lower-lying occupied MOs are found to also contribute to charge transport when the energy levels are close to that of HOMO¹¹. The charge hopping rate constants, k(i^H-p → j^H-q), from the HOMO-p (H-p) of the molecule i to the HOMO-q (H-q) of the molecule j (p, q = 0, 1, 2, …; HOMO-0 (H-0) indicates HOMO (H)) in the MD-constructed amorphous aggregate were calculated based on Marcus theory³⁵:

Here, ({{hslash }}) is the Dirac’s constant, ({k}_{{rm{B}}}) is the Boltzmann constant, T is the absolute temperature, ({q}_{{rm{c}}}) is the charge of the carrier, ({bf{F}}) is the external applied electric field, and ({{bf{r}}}_{{ij}}) is the vector connecting the ith and jth molecules. H(i^H-p ↔ j^H-q) is the electronic coupling between the HOMO-p of the ith molecule and the HOMO-q of the jth molecule, ΔE(i^H-p → j^H-q) is the difference between the energy of HOMO-p of the ith molecule, E(i^H-p), and that of the HOMO-q of the jth molecule, E(j^H-q). The average reorganization energies, 〈λ_ij〉s, for the charge hopping from molecule i to molecule j were calculated to be 0.106 eV for holes, 0.372 eV for electrons (average of 50 molecular pairs) based on Nelsen’s four-point method³⁶. To include the intermolecular effect in λ_ij, we used a quantum mechanics (QM) / molecular mechanics (MM) method^37,38. Using the k(i^H-p → j^H-q) values thus calculated, charge transport kMC simulations³⁹ were performed more than 19,000 times on the above-mentioned MD-constructed amorphous aggregate under the PBC. The simulations were carried out for both holes and electrons with 100 nm transport under the applied voltage between 0.9 and 10 V. The calculated hole and electron mobilities agreed well with those of the time-of-flight (TOF) experiments. To understand the details at the molecular level, microscopic analyses were also performed under the applied voltage of 10 V for several representative carriers and for randomly selected carriers. More computational details are provided in the “Methods” section.

The above simulations provided “distributions” of the mobilities. As shown in Fig. 1, both hole and electron mobilities were found to be broadly distributed with log-normal (log-Gaussian) type, ranging over two orders of magnitude for the 100 nm neat film. The fastest and the slowest hole mobilities were 8.5 × 10^-3 and 4.6 × 10^-5cm²V^-1 s^-1, respectively (the fastest and the slowest electron mobilities were 4.2 × 10^-3 and 3.0 × 10^-5cm²V^-1 s^-1, respectively). To the best of our knowledge, the distribution of charge mobilities is revealed quantitatively for the first time.

a Hole mobilities. b Electron mobilities. The maximum (μ_max) and minimum (μ_min) hole and electron mobilities are shown in the figures (in the unit of cm² V^-1 s^-1). The average (μ_log-log) mobilities obtained from the log-log plots of the computational TOF profiles are also shown.

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Our kMC simulations also provide computational TOF profiles with the same type as experiments. Therefore, commonly determined average hole and electron mobilities can be estimated also in our simulation using the same procedure as in the standard TOF experiments. The hole and electron mobilities thus determined were 4.5 × 10^-3 and 1.6 × 10^-3cm²V^-1 s^-1, respectively (Fig. 1), from the log-log plots of the computational TOF profiles in this study (they were 4.4 × 10^-3 and 1.4 × 10^-3cm²V^-1 s^-1 in the linear-linear plots). The mobilities thus determined are close to those of the fastest hole and electron in the distributions, as clearly shown in Fig. 1, clarifying the existence of carriers with much slower mobilities.

Charge transport between electrodes

To elucidate the details of charge transport, we investigated the elapsed time and the number of hops during the 100-nm transport. Figure 2 shows the results for holes (see Supplementary Fig. 1 for electron transport). From the fastest and the slowest holes (Fig. 2a, b, respectively) and holes with intermediate mobilities (Fig. 2c, d), hole transports are found to consist of two different time regimes: “linear transport” regimes where charges are transported at a constant rate and “temporal trap” regimes where charges stay almost at the same position for a considerable time. The same is true for electron transport.

a The fastest hole. b The slowest hole. c, d Holes with intermediate mobilities. e The fastest hole without n-type trap regimes. f The slowest hole without n-type trap regimes. The details of traps near 93 nm in panel b, near 2 nm in c, and near 15 nm in panel d (indicated by arrows) are described in Supplementary Fig. 3, Figs. 3, and 4, respectively.

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For the fastest hole (Fig. 2a), the elapsed time (red line) and the number of hops (blue line) are found to increase mostly “linearly” during the 100-nm transport. In sharp contrast, those for the slowest hole (Fig. 2b) were found to drastically increase at specific positions, such as ~64 and ~93 nm from the anode, without changing the positions. Except for these specific positions, the time and positional progress of the slowest hole are similar to those of the fastest hole. In this study, frequent charge hoppings forward and backward (“charge catch-balls”) between two or a few specific molecules are called “n-type traps”. Significant time is wasted by the n-type traps at ~64 and ~93 nm in Fig. 2b. To confirm the similarity between the “linear transport” regions of the fastest and the slowest holes, we carried out hypothetical simulations without n-type traps for both the fastest and the slowest holes. The n-type traps are significant for the slowest hole but are also found for the fastest hole, for example, at around 13, 17, 28, 32, 45, and 92 nm from the anode (Fig. 2a). They were also excluded.

Figure 2e, f show the results without n-type traps for the originally fastest and slowest holes, respectively. The number of hops increases linearly with distance for both the fastest and the slowest holes by excluding n-type traps, and have no significant difference between them; the number of hops was 1.8 × 10² for 100-nm transport in both holes. The time required for 100-nm transports was significantly reduced (from 1.2 × 10^-9 to 2.7 × 10^-10s for the originally fastest hole and from 2.2 × 10^-7 to 3.7 × 10^-10s for the originally slowest hole). By excluding n-type traps, considerable elapsed times, such as ~64 and ~93 nm from the anode, disappear, indicating that the n-type trap is one of the origins of retarding charge transport and reducing mobility (this will be named “off-diagonal trap” later). Without n-type traps, the number of hops increases linearly with distance for both the fastest and the slowest holes. However, the elapsed times as a function of distance from the anode, that is, the velocities, are not constant even without the n-type trap during the 100-nm transport (the red lines in Fig. 2e, f). Time wasted at specific locations; for example, at ~7 and ~52 nm in Fig. 2e and at ~21 and ~80 nm in Fig. 2f, indicating that there still exist traps different from n-type traps. Here, we call them “t-type traps” (note that n-type traps are not always dominant; Fig. 2c and Supplementary Fig. 2b are examples where t-type traps are dominant). Without both n-type and t-type traps, the transport behaviors of the originally fastest and slowest holes are indistinguishable.

From the above analysis, we found different types of traps: traps with a large number of hoppings (n-type trap) and traps with a small number of hoppings and large time loss (t-type trap). We have investigated only the fastest and the slowest holes, but to confirm the generality, we also analyzed various other holes. Among them, we show two holes with intermediate mobilities in Fig. 2c, d. Similar to the fastest and the slowest holes, the holes in Fig. 2c, d are also composed of “linear transport” and “temporal trap” regimes. Typical t-type traps are found, for example, at ~2 nm in Fig. 2c, and n-type traps are found at ~15 nm in Fig. 2d. We carefully investigated whether permanently trapped carriers exist or not for all holes and electrons, resulting in all the charges finally reaching the counter electrode at least under the computational condition in this study. Therefore, all the traps here are “temporal traps.” “Permanent traps” may exist under much lower electric fields, F, or for doped films with significant energy gaps.

Molecular-level charge hopping: clarification of “diagonal trap (energetic trap)”

Here, we investigate the molecular-level details of a t-type trap, at ~2 nm in Fig. 2c. This type of trap is generally observed; other examples are found in Supplementary Fig. 2a and b at ~20 nm and ~48 nm, respectively (the details of these t-type traps are shown in Supplementary Tables 1 and 2). Figure 3a shows the actual intermolecular packing (distance and orientation) of the amorphous film at around 2 nm in Fig. 2c, where the molecules are used for hole hoppings at least once. The hash mark represents the molecule number, from #1 to #4000. To show the hopping (charge transfer) rate constants k(i^H-p → j^H-q) and the number of hoppings n(i^H-p → j^H-q) between respective molecular pairs, the intermolecular distances are enlarged in Fig. 3b. Figure 3c shows the energy level E(i^H-p) relative to the HOMO of the site #515 (E(#515^H)), the lowest in energy for holes among all the sites shown here. Although HOMOs were used for all molecules in Fig. 3, HOMO–1’s were used only for molecules #515 and #1270. Figure 3d (Table in Fig. 3) shows the hopping distance along F from molecule #i to molecule #j (Δx(i → j)), ΔE(i^H-p → j^H-q), |H(i^H-p ↔ j^H-q)|, k(i^H-p → j^H-q) and n(i^H-p → j^H-q). For example, the first row is for hole hopping from HOMO of molecule #388 to that of molecule #515; the distance in the direction of F, Δx(#388 → #515), is 6.4 Å, the energy difference, ΔE(#388^H → #515^H), is -0.128 eV, the electronic coupling, |H(#388^H ↔ #515^H)|, is 2.6 meV, and the hopping rate constant, k(#388^H → #515^H), is 1.8 × 10¹¹s^-1. The number of hoppings of one hole from the HOMO of molecule #388 to that of molecule #515, n(388^H → 515^H), is 9. The corresponding backward charge hopping in the opposite direction, from the HOMO of molecule #515 to that of molecule #388, is provided in the third row.

A hole is injected into molecule #388 and finally drained from molecule #1270. a Molecular packing structure near the trap site. b Schematic of intermolecular hopping. Molecular conformations and orientations are correctly represented, but intermolecular distances are adjusted for clarity. k(i^H → j^H) and n(i^H → j^H) are also shown (hoppings using HOMO-1 are omitted due to space limitations). c Energy levels relative to #515^H as a function of distance from the anode. H means HOMO and H-1 means HOMO-1. The energy levels are displayed upside down here. d Charge hopping distances (Δx(i → j)), site energy differences (ΔE(i^H-p → j^H-q)), electronic couplings (|H(i^H-p ↔ j^H-q)|), charge transfer rate constants (k(i^H-p → j^H-q)), and the number of hoppings (n(i^H-p → j^H-q)) for respective molecular pairs. Blue bold pairs are the origin of the traps.

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In Fig. 3d, we found that k(#515^H → #388^H) and k(#515^H → #503^H), that is k(i^H-p → j^H-q)s for hole hoppings from #515 to #388 and from #515 to #503 (all for HOMO), are more than nearly two orders of magnitude smaller than the others, exhibiting that these two time-consuming hoppings are the origin of t-type trap here (the values of n are equivalent to the others). In contrast to the time-consuming forward hopping (Δx(#515 → #503) is positive) with k(#515^H → #503^H) of 1.1 × 10⁸s^-1, the corresponding backward hopping from #503 to #515 in the 7th row (the hopping against F; Δx(#503 → #515) is negative) shows far larger k(#503^H → #515^H) of 8.6 × 10¹⁰s^-1, although the |H(#515^H ↔ #503^H)| is the same (1.6 meV) and n’s are nearly the same (5 and 4) for hoppings in both directions. The reason why an extremely long time is needed for the hole hopping from #515^H to #503^H is the energy difference between the two sites; the hole should overcome the ΔE(#515^H → #503^H) of 0.22 eV, the largest energy gap in Fig. 3. We also found hoppings from HOMO–1 of #515 to HOMO of #503 (the 6th row). Although |H(#515^H-1 ↔ #503^H)| is only 1.8 times larger than |H(#515^H ↔ #503^H)|, and both Δx(515^H-1 → #503^H) and Δx(515^H → #503^H) are the same, k(#515^H-1 → #503^H) of 2.0×10¹¹s^-1 is far larger than k(#515^H → #503^H) of 1.1×10⁸s^-1, confirming that the origin of the small k(#503^H → #515^H) is the energy gap (ΔE(#515^H-1 → #503^H) is only 0.03 eV).

The origin of the temporal trap for hole hopping from #515^H to #503^H is found to be the energy difference. We have named it “diagonal trap (energetic trap).” Table 1 shows the characteristic features. Large ΔE(i^H-p → j^H-q) is the origin of small k(i^H-p → j^H-q). The values of n(i^H-p → j^H-q)’s are small (typically, n < ~10), as exemplified in Fig. 3. We found that even when a carrier drops to an energetically low site, they hop back to the original site in all cases as far as we have investigated, resulting in similar n values between forward and backward hoppings. The value of |H(i^H-p ↔ j^H-q)| is one or two orders of magnitude smaller than the largest |H(#3331^H-3 ↔ #2854^H-2)| of 130 meV (the largest for HOMO-HOMO hopping is |H(#2569^H ↔ #348^H)| of 45.8 meV). The k(i^H-p → j^H-q) values are largely different between forward and backward hoppings due to the large energy difference, depending on whether the hopping is uphill or downhill (one-way trap).

Molecular-level charge hopping: clarification of “backward hopping trap”

The hole at #515 has another site to hop; #388 (Fig. 3). Opposite to the hops between #515 and #503, the forward hopping is very fast, and backward hopping is time-consuming between #515 and #388; in contrast, to the fast forward hopping with k(#388^H → #515^H) of 1.8×10¹¹s^-1, the corresponding backward hopping shows about three orders of magnitude smaller k(#515^H → #388^H) of 1.0 × 10⁸s^-1, although the |H(#388^H ↔ #515^H)| is the same and n(i^H-p → j^H-q)’s are nearly the same (9 and 8). The origin of the extremely small value of k(#515^H → #388^H) is caused by hopping in the opposite direction with respect to F. The hole should overcome the Δx(#515 → #388) of -0.64 nm, the largest negative distance in Fig. 3. We here named it “backward hopping trap.” The energy gap between the two sites, ΔE(#515^H → #388^H) of 0.13 eV, will also contribute to the hopping delay. However, the contribution would be small, because k(i^H-p → j^H-q)s are large enough even when the energy gap is at the same level for forward hopping (see hoppings from #1506^H to #1270^H, from #1270^H to #1034^H in Fig. 3d, from #871^H-1 to #2478^H in Fig. 4d, from #3742^H to #2187^H-1 in Supplementary Fig. 3d, and from #98^H−1 to #1517^H in Supplementary Table 4). The characteristic features of the “backward hopping trap” are also summarized in Table 1. Negative Δx(i^H-p → j^H-q) is the origin of small k(i^H-p → j^H-q). The values of n(i^H-p → j^H-q)s are normally small for both forward and backward hoppings. The values of |H(i^H-p ↔ j^H-q)| are not significant. This is also a one-way trap; the corresponding forward hopping (the 1st row of Fig. 3d) provides far larger k(i^H-p → j^H-q) compared to the backward hopping.

A hole is injected into molecule #2478 and finally drained from molecule #871. a Molecular packing structure near the trap site. b Schematic of intermolecular hopping. Molecular conformations and orientations are correctly represented, but intermolecular distances are adjusted for clarity. k(i^H → j^H) and n(i^H → j^H) are also shown (hoppings using HOMO-1 are omitted). c Energy levels relative to #871^H as a function of distance from the anode. H means HOMO and H-1 means HOMO-1. The energy levels are displayed upside down here. d Δx(i → j), ΔE(i^H-p → j^H-q), |H(i^H-p ↔ j^H-q)|, k(i^H-p → j^H-q), and n(i^H-p → j^H-q) for respective molecular pairs. Blue bold pairs are the origin of the traps.

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Hoppings with small k(i
^H-p → j
^H-q)

The hoppings from HOMO of #515 to those of #503 and #388 occur despite the small k(#515^H → #388^H) and k(#515^H → #503^H) values of the order of 10⁸s^-1, because the k(#515^H-p → j^H-q)’s from HOMO of #515 to all the other sites are far smaller (see Supplementary Table 5). In fact, no hoppings were found from #515 to all sites except to #388 and #503. The third largest k(#515^H-p → j^H-q) next to k(#515^H → #388^H) and k(#515^H → #503^H) is k(#515^H → #43^H) of 2.0×10⁶s^-1, two orders of magnitude smaller, due to the larger ΔE(#515^H → #43^H) of 0.298 eV, although their |H(#515^H↔j^H-q)|s are similar and Δx(#515 → #43) is positive. Sites other than #388 and #503 were energetically much higher in energy, which inhibit hoppings to these sites (n(#515^H → #43^H) = n(#515^H → #2674^H) = n(#515^H → #172^H) = 0).

Molecular-level charge hopping: clarification of “off-diagonal trap (structural trap)”

Another type of charge trap (n-type trap) is found at ~15 nm in Fig. 2d. This type of trap is also generally observed; other examples are found in Fig. 2b at ~93 nm, Supplementary Fig. 2c, d, both at ~41 nm (the details of these n-type traps are shown in Supplementary Fig. 3 and Supplementary Tables 3 and 4). Here, we investigate the molecular-level detail of this n-type trap. Figure 4a shows the local structure of the amorphous film around 15 nm with actual molecular conformations and packings; the molecules are used for hole hoppings at least once. The intermolecular distances are enlarged in Fig. 4b to show the intermolecular hoppings for respective pairs. The values of k(i^H-p → j^H-q) and n(i^H-p → j^H-q) are also shown. Figure 4c shows their energy levels. Figure 4d shows the values of Δx(i → j), ΔE(i^H-p → j^H-q), |H(i^H-p ↔ j^H-q)|, k(i^H-p → j^H-q) and n(i^H-p → j^H-q) for respective molecular pairs, similar to Fig. 3.

The origin of the n-type trap is hoppings between #871 and #1896, clearly different from the above t-type traps. Contrary to the case of “diagonal trap” and “backward hopping trap” in Fig. 3, where forward hoppings and backward hoppings have significantly different k(i^H-p → j^H-q) values (k(#388^H → #515^H) >> k(#515^H → #388^H) and k(#515^H → #503^H) << k(#503^H → #515^H)), both the forward k(#1896^H → #871^H) and the backward k(#871^H → #1896^H) are of similar magnitude (Fig. 4d). In the case of “diagonal trap” and “backward hopping trap” in Fig. 3, k(i^H-p → j^H-q)’s of the traps (k(#515^H → #388^H) and k(#515^H → #503^H) of ~10⁸s^‒1) were much smaller than those of the others (~10¹⁰–10¹²s^‒1). However, n-type traps in Fig. 4 exhibit the opposite feature compared to the t-type traps; both k(#871^H → #1896^H) and k(#1896^H → #871^H) (~10¹³s^‒1) are larger than or at least the same order as those of the surroundings. Normally, when the elementary hopping is slow (k(i^H-p → j^H-q) value is small), the site becomes a trap. In contrast, the k(#1896^H → #871^H) and k(#871^H → #1896^H) values are large but become traps, because n(#1896^H → #871^H) and n(#871^H → #1896^H) are far larger than the others (Fig. 4d). The two sites are close in space (|Δx(#1896 ↔ #871)| = 0.23 nm), the electronic coupling (|H(#1896^H ↔ #871^H)|(,=) 25 meV) is larger than those of the surroundings, and the energy difference is nearly zero (|ΔE(#1896^H ↔ #871^H) | = 0.001 eV), resulting in large hopping rate constants for both the forward and backward hoppings. The large k(#1896^H → #871^H) and large k(#871^H → #1896^H) compared to the surroundings result in repeated forward and backward hoppings between the two molecules (n(#1896^H → #871^H) and n(#871^H → #1896^H) are ~5400 for both). These features of the n-type trap are completely different from the above-mentioned “diagonal trap” and “backward hopping trap”; short Δx(i → j), large |H(i^H-p ↔ j^H-q)|, and small |ΔE(i^H-p → j^H-q)| (Table 1). Then, the values of k(i^H-p → j^H-q)’s become large and similar order of magnitude both for forward and backward hoppings, resulting in very large n(i^H-p → j^H-q)’s for hoppings in both directions. Since both the short |Δx(i → j)| and large |H(i^H-p ↔ j^H-q)| compared to the surroundings originate from the off-diagonal disorder, we named it “off-diagonal trap (structural trap).” Normally, it is considered to be preferable to shorten the intermolecular distance and increase the electronic coupling to increase charge mobility, and such molecular designs have been performed. However, this example here shows a finding that a pair with “locally” large |H(i^H-p ↔ j^H-q)| can be a trap when the surrounding |H(i^H-p ↔ j^H-q)|’s are small. This occurs in the case of amorphous systems.

Molecular-level charge hopping: combined and multiple traps

We also found combined traps of the three types in this study. At ~93 nm in Fig. 2b (see also Supplementary Fig. 3), the charge is trapped because both n(2187^H → 3143^H) and n(3143^H → 2187^H) are very large (>3900). Of these hoppings, Δx(#3143^H → #2187^H) of −0.68 nm also makes it a backward hopping trap. The repeated fast forward and slow backward hoppings due to the combined off-diagonal trap and backward hopping trap retard the carrier significantly. This trap site at ~93 nm contains a backward hopping and off-diagonal combined trap (#3143^H → #2187^H), off-diagonal traps (#2187^H → #3143^H and #3742^H ↔ #2187^H), a backward hopping and diagonal combined trap (#3143^H → #2187^H-1) and a diagonal trap (#3143^H → #3644^H). These combined and multiple traps result in the slowest hole mobility among all the carriers.

In this study, we have performed a detailed molecular-level analysis of charge transport in an amorphous aggregate based on a multiscale simulation, which has been difficult to access so far. The analysis reveals three types of traps, “diagonal trap,” “backward hopping trap,” and “off-diagonal trap,” all of which control the charge transport. They are all temporal traps; permanent traps (completely trapped immobile charges) have not been found in our simulation conditions. A molecular pair with k(i^H-p → j^H-q) smaller than the surroundings becomes a “diagonal trap” or “backward hopping trap,” depending on the origins. The “diagonal trap” originates from the large energy gap between the hopping sites, ΔE(i^H-p → j^H-q), providing a small k(i^H-p → j^H-q). It takes considerable time to hop from a low-energy site to a neighboring site; therefore, this trap can also be called “energetic trap.” Another origin of small k(i^H-p → j^H-q) is charge hopping against the direction of the applied electric field. It takes considerable time to hop against the applied electric field; we named it “backward hopping trap.” This is different from the “diagonal trap” because it occurs even when ΔE(i^H-p → j^H-q) is small. These two types are one-way traps; it takes a long time for i → j hopping, but the opposite j → i hopping is fast. The “off-diagonal trap” is completely different. Due to the locally large |H(i^H-p ↔ j^H-q)|, and therefore, large k(i^H-p → j^H-q) both from i-site to j-site and from j-site to i-site, charges hop back and forth frequently only between the two molecules, resulting in wasting of time. Since locally small |Δx(i^H-p ↔ j^H-q)|s and therefore locally large |H(i^H-p ↔ j^H-q)|s in the distributed structures are the origin of traps, this can also be called “structural trap.” Different from a molecular aggregate exhibiting high mobility with a series of continuously large |H(i^H-p ↔ j^H-q)| like single crystals, a molecular pair with |H(i^H-p ↔ j^H-q)| locally larger than the surroundings conversely decreases mobility. This occurs in disordered aggregates, like amorphous solids. It is normally considered that larger |H(i^H-p ↔ j^H-q)| and larger k(i^H-p → j^H-q) is favorable to achieve faster mobility. However, our simulation in this study clearly shows that carriers are hard to escape from the pair with locally large |H(i^H-p ↔ j^H-q)| in disordered systems. This has a negative effect on increasing charge mobility when the |H(i^H-p ↔ j^H-q)| is far larger than those of surroundings.

In amorphous molecular systems, both diagonal disorder and off-diagonal disorder are present originating from the structural disorder. In addition, hopping against bias is also commonly observed. Traps based on these will be found in any kind of amorphous molecular system; therefore, the findings of this study can be generalized to a wide range of new as well as existing amorphous molecular systems.

Multiscale simulation

The multiscale simulation was performed as follows. The molecular structure of an isolated CBP was optimized by a DFT method using B3LYP functional and 6-31 G* basis set. Using the DFT-optimized CBP molecule as an initial structure, an MD simulation was carried out to construct amorphous aggregates consisting of 4000 molecules in total under the PBC. The MD simulation was first performed in the NVT ensemble at 573 K to mimic the experimental vacuum deposition temperature and then in the NPT ensemble at 300 K. In the MD simulation, the Dreiding force field⁴⁰ was used, where the equilibrium bond length and angle parameters of the force field were modified using those of the DFT-optimized molecule. The charge hopping rate constants, k(i^H-p → j^H-q)’s, in the MD-constructed amorphous aggregate were calculated at 300 K based on Marcus theory³⁵ in Eq. (1). The value of ΔE(i^H-p → j^H-q) is positive for endothermic hopping and negative for exothermic hopping for hole transport. For the QM/MM calculations of reorganization energy, ({lambda }_{{ij}}), DFT (B3LYP/6-31 G*) and the Dreiding force field were used for the QM and MM regions, respectively. We consider molecules within 3.0 nm for the MM region, and their geometries were frozen during the calculations. Since the DFT calculations of λ_ij for all the molecular pairs in the aggregates were impossible due to the high calculation cost, we used an average λ_ij, 〈λ_ij〉, using randomly selected 50 molecules. The values of 〈λ_ij〉’s were 0.106 eV for holes and 0.372 eV for electrons. Using the k(i^H-p → j^H-q) values thus calculated, charge transport kMC simulations were performed at 300 K. The DFT and the QM/MM calculations were performed by Gaussian 16 program package⁴¹. MD simulation was conducted by the LAMMPS program package⁴². The kMC simulations were run by our in-house code, written in Python and C + +.