Density of States and the Role of Energetic Disorder in Charge

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Density of States and the Role of Energetic Disorder in Charge Transport in an Organic Radical Polymer in the Solid-State Travis W, Kemper, Ross E. Larsen, and Thomas Gennett J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b06368 • Publication Date (Web): 27 Aug 2015 Downloaded from http://pubs.acs.org on September 4, 2015

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The Journal of Physical Chemistry

Density of States and the Role of Energetic Disorder in Charge Transport in an Organic Radical Polymer in the Solid-State Travis W. Kemper,† Ross E. Larsen,∗,† and Thomas Gennett‡ Computational Science Center National Renewable Energy Laboratory 15013 Denver West Parkway, Golden CO 80401, and Chemical and Nano Sciences Center E-mail: [email protected]

Phone: (303) 275-4422 . KEYWORDS: organic, electronic, battery, nitroxide, modeling, electron-transfer

∗ To

whom correspondence should be addressed Science Center National Renewable Energy Laboratory 15013 Denver West Parkway, Golden CO 80401 ‡ Chemical and Nano Sciences Center † Computational

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Abstract On the basis of atomistic simulations of the stable organic radical polymer material, poly (2,2,6,6-tetramethylpiperidinyloxy methacrylate) (PTMA), various material properties relating to charge transport were evaluated in terms of the Marcus charge transfer rates between radical sites. The reorganization energy of the PTMA monomer unit was calculated using DFT to provide an approximate value to enter into the Marcus charge transfer rate. The role of energetic disorder in the charge transfer between sites caused by the different local environments seen by radical sites is examined in terms of both steric and electrostatic effects. The electronic coupling between sites was examined in terms of the inter-site network, morphological features and energetic disorder. Energetic disorder was found to result in both sites that act as traps and paired sites that were highly coupled to each other and would act as a single site for transport purposes.

1 Introduction Electronic materials based on organic semiconducting molecules and polymers, represent one avenue in the future development of energy harvesting and energy storage. Organic materials offer the possibility of environmentally friendly, low cost and recyclable electronics that can be made flexible, stretchable and wearable. 1,2 In addition, organic electronic materials allow for scalable processing methods such as roll to roll processing and inkjet printing that have low capital expenditure requirements compared to traditional inorganic electronic materials such as silicon. 3,4 While, the field of organic electronic materials represents a new and exciting path for electronic devices, there still exist large barriers to them being implemented as commercial devices. These barriers include finding materials that possess desirable properties, ensuring consistent device properties during fabrication, lifetime and reliability. 4 Underlying all of these barriers is an interplay between molecular properties (e.g., chemical and electronic structure) and material properties (e.g., inter-molecular coupling and film morphology). Hence, efforts to overcome the aforementioned hurdles have involved identifying the effects of intra-molecular interactions and relating 2 ACS Paragon Plus Environment

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and via this continuous redox process PTMA can be electrically conductive. In this work we analyzed the rate of charge transfer between the pendant group sites according to Marcus theory and have identified factors that can lead to variations in the charge transfer rate kCT . In a previous paper, 13 we calculated the electronic coupling between nitroxide radical sites and pointed out that two primary inter-site packing motifs that contribute to the diffusion of holes through the material: (a) “head on” configurations with a relatively large (100 meV ) electronic coupling associated with a close contact inter-nitroxyl nitrogen distance of 4.5 Å; and (b) parallel nitrogen-oxygen bond stacking configurations with an electronic coupling of approximately 10 meV at an inter-nitroxyl nitrogen distance of 6.5 Å. Our analysis led us to conclude that absent inter-site energetic disorder, fully 85% of the charge transfer events occur between sites on different polymer chains, rather than involving charge transfer along a single chain. Here we continue this analysis of the other terms that enter the Marcus charge transfer rate, with the focus on the role played by site energy disorder in modifying charge transfer rates in terms of fundamental molecular and morphological properties. We computed the density of states (DOS) of the ionization potential of the TEMPO groups as modified by the local environment in a simulated film, including steric and electrostatic effects. These site energies are used to examine how energetic disorder affects charge transfer rates in PTMA. We identify two sub-sets of sites, nearlyisolated sites and paired sites that may act as traps to reduce charge mobility. The implications of the energy dispersion and trapping sites have for charge conduction in PTMA is discussed.

2 Approach and Computational Methodology As we established previously, 13 the electronic orbitals associated with the radical are localized to the vicinity of the nitroxide N-O bond, Figure 1, not in delocalized bands. Therefore, a hoppingmodel for transport based on the Marcus theory is appropriate. In order to relate material properties to charge transfer rates and the resulting hole mobility we must consider all the terms in the Marcus charge transfer rate and calculate rates between sites. The mobility of a material with localized


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charge carrier sites is determined both by the rate at which a charge carriers, either holes or electrons, can transfer from site to site and by the connectivity of the localized charge site network. Holes diffuse through the material based on a homogeneous oxidation/reduction reaction occurring between sites i and j, i+ + j• → i• + j+ .

(1)

During this process an electron is transferred from the singly occupied molecular orbital (SOMO) of a radical site ( j• ) to the lowest unoccupied molecular orbital (LUMO) of the cation site (i+ ). The rate of charge transfer between sites is given by the Marcus rate,

kCT =

Vi2j h¯

π kB T λ

1/2

"

exp −

(λ + ∆G◦i j )2 4 λ kB T

#

,

(2)

where T is temperature, Vi j is the electronic-coupling matrix element between the sites i and j, λ is the reorganization energy, ∆G◦i j is the change in free energy at standard pressure and temperature associated with the charge-transfer reaction and h¯ and kB are Planck’s constant over 2π and Boltzmann’s constant, respectively. 14,15 It is the contribution of each of the material-specific components (Vi j , λ and ∆G◦i j ) to the network of charge transfer rates in a PTMA film that let us infer design rules for better conducting organic radical materials. It has been pointed out recently 16 that if molecular motions modulate the inter-site coupling on timescales approaching those for intersite electron transfer, one should replace Vi2j with its averaged value, hVi2j i. Because of the strong steric hindrances for site motions within a PTMA film, we do not expect this to be needed here. The film morphology examined here was produced as reported in a previous paper, 13 using the GROMACS molecular dynamics (MD) code and using the AMBER force field (FF) modified for radical structures 17–19 on 101 oligomers of syndiotactic PTMA, which were 24 monomer units long. The syndiotactic configuration implies that the methyl and ester groups coming fof the backbone should be rotated 180 degrees for alternating units but for simplicity we have elected to show only one orientation in the repeat unit of 1. As described there, inter-site electronic couplings 5 ACS Paragon Plus Environment


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were computed for FF-minimized structures using NWCHEM at the Hartree-Fock (HF) level with a 3-21G basis set, in the gas phase 20,21 for the 12, 026 pairs of TEMPO groups with inter nitroxide nitrogen separations less than 10 Å. We utilized these previously computed values of Vi j in our current analysis. Within the film environment, steric and electrostatic effects can cause the energy of a hole to vary from site to site, thus leading to a nonzero values of ∆G◦i j for the nominally homogeneous hole transfer reaction. Our focus is on hole transfer, so the energies we calculate for a hole on each site should correspond to the ionization potential (IP) of that site, as the IP reflects how much energy is need to remove an electron from the SOMO level. The FF used in the simulation does not have parameters for the geometry of a TEMPO cation, so the cation geometry could not be captured within our current model. Accordingly, the vertical IP (vIP) was calculated using the geometry of the neutral radical TEMPO group in the film. Specifically, site energies were computed using two single point gas phase energies for each of the 2, 424 FF-minimized TEMPO groups in the neutral geometry (ng): the energy of the neutral site, E(ng)• , and the cation energy in the neutral geometry E(ng)+ . The difference is the vIP, vIP = E(ng)+ − E(ng)• .

(3)

Furthermore to account for the electrostatic effects on cation energies due to the electrostatic environment of the film in the surrounding film a local electric potential Φi was calculated based on the atomic charges of the surrounding medium,

Φi =

1 qn ∑ 4πεεo n6=i |r~n−i |

(4)

where εo is the permittivity of free space, ε was taken to be 3.4 from capacitive-voltage measurements 22 and qn is the atomic charge of atom n, defined for the PTMA monomer in our previous work based on an electrostatic potential (ESP) fit to the HF results. 13 Note that the only atomic sites, n, included in the sum are those that are not in the set of atoms that comprise site i (which 6 ACS Paragon Plus Environment

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is all the atoms in the TEMPO group at site i, as described in our previous paper). 13 The distance between the atom n, and the site, i, (|r~i−n |) was approximated by taking the distance of atom n to the nitroxyl nitrogen of the TEMPO group i. The shift in site energy due to the local electric potential (LEP) was found by adding the local electric potential to the energy of the hole vIPLEP = (E(ng)+ + qΦ) − E(ng)•

(5)

where q is the charge of a hole since we are considering positively charged carriers, i.e. the oxoammonium cation. These quantities allow us to define site intrinsic energies for each of the TEMPO groups in the film based on the combination of conformational/steric (vIP) and electrostatic (Φ) effects. Based upon these site energies a change in energy for a hole to move from site i to site j can be defined. Because the hole transfer is from TEMPO group to TEMPO group, we anticipate that the entropic contribution to the change in Gibbs free energy should be negligible, so we approximated the free energy difference, ∆G◦i j , as the change in enthalpy ∆Hi j ∆Hi j = [vIPLEP ( j)] − [vIPLEP (i)].

(6)

The final term of interest in understanding hole transport in PTMA is the reorganization energy. The reorganization energy can be divided into two components, the first is the geometric relaxation of the reacting molecules, i+ upon being reduced and j• upon being oxidized, which is referred to as the inner-shell reorganization energy λ inner . The second is the change in energy of the surrounding environment in response to the movement of the hole from i to j, which is referred to as the outer-shell relaxation energy λ outer . In this work, we computed only the inner-shell reorganization energy. For site i in Reaction 1 the molecule starts in the cation state, then is reduced to the neutral charge state in the cation geometry (cg) with an energy of E(cg)• and relaxes to the neutral radical geometry (ng), with an energy of E(ng)• . Similarly, site j begins in the relaxed neutral state and is oxidized to the cation charge state with an energy of E(ng)+ , then relaxes to 7 ACS Paragon Plus Environment


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the cation geometry with an energy of E(cg)+ . 23 The inner shell reorganization energy, λ inner , is the total energy difference associated with these relaxations for molecules i and j, ∆H • = E(cg)• − E(ng)•

(7)

∆H + = E(ng)+ − E(cg)+

(8)

λ inner = ∆H + + ∆H • .

(9)

Given this definition of the Marcus charge transfer rate, Equation 2, we calculated the charge transfer rates between sites in a simulated PTMA film, and conduct analysis on each term to determine how it effects the mobility of PTMA.

3 Results 3.1 Site energy distributions First we will examine the density of states (DOS) of the SOMO level within the film. The DOS (NSOMO (E)) is defined as the number of SOMO states per unit volume per unit energy at energy E. PTMA is not an intrinsic semiconductor so the probability of the SOMO being occupied at equilibrium is unity. Consequently integrating NSOMO (E) f (E)dE, where f (E) is the probability of occupancy, over all energies will simply give the number density of radicals per unit volume, which was found in our simulations to be 264×1019 cm−3 . In this work we are primarily interested in how different components of the PTMA film affect the DOS. First we considered the steric distortions imposed on each TEMPO group due to its surrounding neighbors and the effect of these surrounding-induced TEMPO geometry changes is shown in DOS of the vIP in Figure 2. The SOMO DOS we computed was approximately two orders of magnitude smaller than the full DOS of the conjugated material P3HT 24 due to the lower concentration of radical sites and the fact that we include only a single state rather than many electronic states per monomer unit. The width (2σ , see Equation 10) of this DOS was found to be 160 meV with an average value of 7.46 eV . 8 ACS Paragon Plus Environment

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While the average value differs from the reported experimental value of the SOMO level, which ranges from 5.2 eV to 5.4 eV 22,25,26 it is only the difference in the vIP values that enters into the charge transfer rate, Equation 6. Therefore, while a relatively low level of theory (HF/3 − 12g) was used to perform the 4, 848 calculations needed to compute the DOS, the trends were well captured and gave reasonable values of ∆H. Including the local electronic potential, Φ, in the calculations of the DOS (vIPLEP ) caused ¯ a decrease in the average value to 7.35 eV , Figure 2, because Φ was −0.38 V on average (Φ). ¯ was due to both the large dipole moment of TEMPO, which was previThe negative value of Φ ously calculated to be 3.24 Debye, 13 and the fact that in the simulated film each radical site has a negatively-charged oxygen atom pointing towards it. Hence, on average holes are stabilized in PTMA. That is, it is easier to ionize TEMPO radicals in the film than in isolation. The electric potential at each TEMPO site also increased the variation of hole energies leading to a broadening of the DOS from 160 meV to 240 meV. Taking the dielectric constant to be the bulk value of 3.4 may not be strictly valid on such small length scales so we also considered the case of a dielectric constant of 1.0, giving a range of electrostatic site energies with and without dielectric screening. This has the added advantage of probing the electrostatic effects to the maximum range, better elucidating their effects on the charge transfer rate. For the case of an ε of 1.0 the average value of the DOS was found to shift to 7.08 eV and the width broadened to 660 meV as the values electric potential were simply increased by a factor of 3.4. The energetic distribution of the SOMO in organic systems is expected to be normal, 27–30 so we plotted a Gaussian function

D(E) = Dmax exp(−

¯ 2 (E − E) ) 2σ 2

(10)

where Dmax is the maximum value of the distribution, E¯ is the average energy, and σ is the standard deviation of the energy for each DOS shown in Figure 2. To report D(E) as a volume energy density, we scaled the normalized energy density of states per site by the number density of TEMPO




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energy, λ . The rate of charge transfer is modulated both by the difference between λ and the inter-site energy differences (∆Goij ), and by the magnitude of λ itself. The larger the reorganization energy, the slower charge transfer. To estimate λ , gas phase calculations were performed on a PTMA monomer to establish the inner-shell reorganization energy (λ inner ). The methyl terminated monomer was optimized in the neutral radical state to get the energy E(ng)• and in the cation singlet state for E(cg)+ , at the B3LYP/cc-pVTZ level. Single point calculations, also at the B3LYP/cc-pVTZ level, were then conducted of the cation ground state geometry in the neutral radical electronic state (E(cg)• ) and the neutral radical geometry in the cation electronic state ( E(ng)+ ). Using Equation 9, ∆H + was found to be 0.552 eV and ∆H • was found to be 0.475 eV . This gives an approximate gas phase inner-shell reorganization energy of (λ inner ) of 1.0 eV. Given the uncertain value of the outer-shell reorganization energy we used 1.0 eV as a baseline, and examine charge transfer rates for a range of reorganization energies.

3.2 Inter-site transfer rates The values of kCT were calculated for each pair i j and ji within 10 Å of each other in the MD snapshot. We computed all of these kCT values for temperatures ranging from −20 C to 80 C, for reorganization energies λ ranging from 0.5 eV to 1.5 eV , and with vIP calculated with an ε of 3.4 and with ε = 1. The probability densities of kCT as functions of these variables are shown in Figure 4. Changing λ across the range mentioned, Fig. 4(a), can shift kCT by several orders of magnitude, as one would expect from λ ’s presence in the exponential term in the Marcus charge transfer rate equation, Equation 2. Next the variation of kCT was investigated for different ways of computing site energies, changing ∆G◦i j . By setting ∆G◦i j to zero a base distribution was calculated using a λ of 1.0 eV and a temperature of 27 C, Figure 4(b). When only the steric disorder is included in the site energies, the probability of finding larger values of kCT increased, because the larger negative values of ∆Hi j reduce the exponential term. Including electrostatic effects in the site energy disorder, which further increased the breadth of the site energy distribution allows even larger negative values of ∆Hi j . Finally, the effect of temperature on the distribution of kCT was 12 ACS Paragon Plus Environment

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preventing hole motion on time scales that might be relevant to a given measurement of charge transport. To estimate a device-relevant time scale, the case of a transport measurement based on applying an electric potential, φ , to sweep charges with mobility, µ , out of a layer of thickness, d was considered. On average, the time for a charge to be extracted from the layer would be hti = d 2 /(φ µ ). Charges that occupy sites for longer than hti can be considered charge traps in such a measurement. The discussion of devices above that we used to estimate lifetimes to define traps does not indicate that we are considering devices in this work: all of our calculations are for intrinsic properties of PTMA films with no reference to electrodes and the attendant interface effects that can dominate device performance for organic semiconductor devices. For PTMA, to estimate what fraction of sites were traps according to the above definition as a function of temperature and reorganization energy we considered a device of thickness 100 nm with a 10 V potential applied and with a nominal mobility of 1x10−4 cm2V −1 s−1 . 22 The large applied potential makes this a conservatively large estimate for the critical trapping rate, kcrit = 1/hti = 1 × 107 s−1 . The rate at which a charge can leave a site, i, was given by the sum of ET rates to all other sites, ktot = ∑ j6=i kiET j , with ki j given by Equation 2. In Table 1, we show the

percentage of trap states, ptrap , and ρtrap associated number density of trap states in the simulated film, as a function of temperature, reorganization energy, using our nominal value for kcrit , and computing the inter-site enthalpy differences with a dielectric constant of 3.4 as discussed above. The results indicate that the percentage of traps are relatively small, only 3%, for room temperature and λ equal to 1.0 eV. 3% is high enough, however, that we might reasonably expect a fraction of holes moving through the film to become trapped on a microsecond time scale. This could lead to charging, which would then affect the mobility. The strong dependence of the trap percentage on the temperature and reorganization energy seen in the table suggests that trap sites are not TEMPO groups that are isolated by small electronic coupling, Vi j , to neighbors. Rather, the traps are sites with energies well below the energies of their neighbors, a conclusion which is supported by the nearly exponential dependence of the tabulated trap density with inverse temperature. In other words, the traps were controlled by the local energetic disorder rather than electronic coupling.


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Table 1: Percentage of traps and paired sites in a PTMA film: percentage of traps, ptrap and density of traps, ρtrap , for a 100 nm thick device with 10 V applied and for a nominal charge mobility of 10−4 cm2 /V s, as described in the text; percentage of sites that are members of tightly coupled pairs p pair and density of pairs, ρ pair , as defined in the text.

λ (eV) T (C) 0.5 -20 0.5 27 0.5 80 1.0 -20 1.0 27 1.0 80 1.5 -20 1.5 27 1.5 80

ptrap ρtrap (1019 cm−3 ) 0.29 0.76 0.00 0.00 0.00 0.00 11.92 31.48 3.22 8.50 0.50 1.31 84.61 223.38 57.22 151.06 24.59 64.91

p pair 4.79 4.46 4.21 4.29 4.04 4.13 3.96 3.71 4.13

ρ pair (1019 cm−3 ) 12.63 11.76 11.11 11.33 10.67 10.89 10.46 9.80 10.89

The trap densities given in Table 1 are strongly influenced by the energetic disorder among the sites, shown in Figures 2 and 3. If the transition rates are computed with the site energies all set to have the same value, so that the rates from Equation 2 are proportional to |Vi j|2 e(−λ /4kBT ), then with λ = 1 eV and at T = 27 C, we have roughly 1/15th as many traps as with site energy disorder, ptrap = 0.21 and ρtrap = 0.54cm−3 . Hence, we conclude that site energy disorder and not heterogeneity in inter-site coupling dominates trapping processes. In addition to hole trapping by energetically isolated sites described above, there was another phenomenon associated with the highest inter-site rates that effectively pairs some sites when considering hole transport on large length scales. For tightly coupled sites with large Vi j and values of ∆Go near zero, that a hole can hop rapidly back and forth between the two sites but this rapid motion is unrelated to bulk charge transport. Eventually, of course, such a hole would be able to exit the pair by hopping to another site, but if the time scale to exit a pair was much less than the time scale associated with inter-pair hopping, the pair must be considered a single object for transport purposes. The concept of finding pairs of sites that are most strongly coupled to each other has precedents in studies of structure and relaxation in the liquid state. 31–33 These studies focused on pairs of atoms that are closer to each other than to any other atoms, so-called mutual nearest neighbor pairs. We found that in our simulated film, 1, 458 of the 2, 424 sites (a fraction



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approximately 0.60) were mutual nearest neighbors (mnn) of each other. It has been shown, 32 that for randomly distributed sites one would expect a fraction 16/27 ≃ 0.59 of mutual nearest neighbors, which suggests that the inter-site packing does not involve ordering that was strong enough to reduce this fraction significantly. This then leads to the question: with fully 60% of the sites members of mnn pairs, does this mean that a large fraction of the sites can be eliminated from consideration for charge transport? In order to answer this questions on must think about pairing based on time-scale separations for the inter-site charge transfer rates themselves and define pairs of sites which have each other as the site with the largest value of kCT and for which the next-fastest rate from each of the sites in the pair is small compared to the inter-pair rate. The first condition leads to between 27% and 34% of sites being most strongly coupled to each other for the ranges of parameters given in Table 1. We chose to define a two sites as paired when the rate between the members of the pair is at least 10 times greater than the next largest rate from either of the paired sites. Table 1 shows the percentage of pair sites, p pair , for a variety of temperatures and reorganization energies. The percentage was between 3.7% and 4.8% for all parameters considered, not the 60% found when only spatial separations were considered. Hence, we find that about 4% of the sites in PTMA can be thought of in some sense as a single site, at least as far as true transport of charges through a film is concerned.

4 Discussion In this work we have examined the terms that enter the Marcus charge transfer rate, Equation 2, to understand charge transfer in PTMA. This was accomplished through a combination of classical molecular simulation and quantum electronic structure calculations. We computed vIP for 2, 424 TEMPO groups and Vi2j for 12, 026 TEMPO pairs in a simulated syndiotactic PTMA film. The distribution of kCT was plotted for different values of λ , ∆Goij (≃ ∆Hi j ) and temperature. Larger values of λ significantly decreased the charge transfer rates across the entire distribution. Fur-


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thermore, once we included steric and electrostatic disorder factors, which increased the width of ∆H distribution, the larger negative values of ∆H allowed increased the number of very large rates observed. Changing the temperature by 100 o C was found to have the little effect, corresponding well to reports that the conductivity of PTMA lacks temperature dependence. 22 Comparing the distribution of kCT in Figure 4 to experimental values of TEMPO in solution, which were found to be on the order of 108 M −1 s−1 , 34 some rates were found to substantially larger 1010 s−1 . This is due to the close contact packing of the "head on" configuration forcing the TEMPO groups into closer contact than would occur in solution. Based on simulations and calculations of a neat syndiotactic PTMA film the effects of different materials properties on the charge transfer rate were determined. The molecular geometry of PTMA was found to produce morphologies with well dispersed nitroxide radical sites with a very low trap density. This helps explain how a 100% of sites can be oxidized when PTMA is used as organic radical battery material. 10 Furthermore, the hole traps were due to local energetic disorder rather than weak inter-site electronic coupling. Therefore, although decreasing the energetic disorder would be a path to reduce the trapping of charges in the film, the primary contribution to this energetic disorder were large electrostatic effects attributed the large dipole moment of a the TEMPO group, which is likely an intrinsic property of these materials. Hence, tuning the local electrostatic energy thus would require attention to packing, morphology and processing that goes beyond mere changes to the radical-containing pendant groups. Another route suggested by our calculations is to modify the reorganization energy, since this has the largest effect on the distribution of charge transfer rates. As a fundamental property of the TEMPO group, this should be considered the primary target for decrease in the development of next generation organic radical based electronic materials. In addition to energetically isolated trap states, we also have identified paired sites that undergo such rapid inter-site charge transfer reactions that they can be considered as a single site for transport purposes. The fraction of paired sites is more than ten times smaller than the fraction of mutual nearest neighbors because the hetereogeneity in inter-site coupling renders some mutual-nearest-



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neighbor pairs not-as-strongly coupled to each other as would be dictated purely by geometry. The fact that several percent of sites can be thought of as pairs in these systems may have implications for design of materials based on avoiding these pairings that can reduce the effective density of transport-relevant states.

Acknowledgement This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under contract DE-AC36-08GO28308. The research was performed using resources sponsored by the Department of Energy’s Office of Energy Efficiency and Renewable Energy, located at the National Renewable Energy Laboratory.

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Density of States and the Role of Energetic Disorder in Charge

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