Chemical defects in polyethylene (PE) can deleteriously downgrade its electrical properties and performance. Although these defects usually leave spectroscopic signatures in terms of characteristic luminescence peaks, it is nontrivial to make unambiguous assignments of the peaks to specific defect types. In this work, we go beyond traditional density functional theory calculations to determine intra-defect state transition and charge recombination process derived emission and absorption energies in PE. By calculating the total energy differences of the neutral defect at excited and ground states, the emission energies from intra-defect state transition are obtained, reasonably explaining the photoluminescence peaks in PE. In order to study the luminescence emitted in charge recombination processes, we characterize PE defect levels in terms of thermodynamic and optical charge transition levels that involve total energy calculations of neutral and charged defects. Calculations are performed at several levels of theory including those involving (semi)local and hybrid electron exchange-correlation functionals, and many-body perturbation theory. With these critical elements, the emission energies are computed and further used to clarify and confirm the origins of the observed electroluminescence and thermoluminescence peaks.
Polyethylene (PE) is an important insulation material that has found widespread use in electrical applications including transmission line cables and capacitors.1 The electrical performance of PE over the long term is affected by impurities and chemical defects that are originally part of the material, as well as those that are created progressively with time. Such defects can introduce charge carrier (or defect) states within the band gap of PE, can act as traps and sources of charge carriers, catalyze further damage, and can deleteriously affect the overall conduction behavior of the insulator.2–6 It is thus critical that a firm understanding of the nature of such defects be obtained.
The best evidence for the presence of chemical defects (especially in PE), and a knowledge of their type, is provided by a variety of luminescence measurements, including photoluminescence (PL), electroluminescence (EL), and thermoluminescence (TL).7–10 Defects, depending on their type and the specific details of the energetic placement of their defect levels, lead to characteristic emission signatures. Such optical measurements have led to the identification of a plethora of defects in PE, the chief among them being the carbonyl, dienone, vinyl, hydroxyl, etc., as illustrated in Fig. 1. Nevertheless, assignments of the luminescence emission bands are never straightforward, and an alternate check of whether a particular defect will lead to a particular emission peak or band is highly desired.
Such an alternate route to identify emission signatures can be provided by first principles computations, e.g., those based on density functional theory (DFT).11,12 Indeed, DFT and beyond-DFT computations have played a critical role in successfully unraveling the repercussions and signatures of defects in inorganic semiconductors and insulators in the past.13–15 In the case of PE, such work is in a state of infancy.16–20 Past DFT work on PE is dominated by the use of (semi)local exchange-correlation (XC) functionals to treat the quantum mechanical part of the electron-electron interaction within a single-particle framework. Defect states are identified as the one-electron Kohn-Sham eigenenergies and the emissions are then assumed to associate with the transitions involving these defect states and either the conduction band minimum (CBM) or the valence band maximum (VBM).17–20 While this approach does provide a qualitative picture of the defect-derived energy levels, there are several fundamental drawbacks. The first one is that the one-electron levels of DFT have no physical meaning as they relate to the energy levels of a fictitious set of non-interacting electrons.12 The second issue relates to the band gap problem of traditional DFT.21 It is well-known that the band gap of insulators is significantly underestimated, which place uncertainties on the energetic location of the defect energy levels within the band gap as well as the position of the conduction and valence band edges.22,23 Finally, intra-defect transitions, i.e., those involving the ground and excited states of neutral defects, are not considered. These factors clearly call into question the emission energies derived from such calculations.
In the present work, the energy emitted due to intra-defect state transition is computed by total energy differences of the neutral defect at excited and ground states and used to explain the origins of PL in PE. The defect levels involving charge recombination processes are characterized in terms of thermodynamic and optical charge transition levels that involve total energy differences of charged and neutral defect calculations. It is well-known that total energy differences provide a formally correct and quantitatively better description of energy level differences in molecules and solids (including defect level placements).14,15,24 Exchange-correlation functionals more advanced than the (semi)local ones used in the past are considered in the present work to critically assess the role of the level of theory adopted. Furthermore, many-body perturbation theory computations are undertaken to accurately determine the band gap of PE and its conduction and valence band edge positions.
The combination of computations undertaken here including the determination and usage of charge transition levels (rather than the one-electron levels) and energy differences of the defect at higher and lowest excited states allows us to directly and quantitatively connect with available luminescence data for PE. The present work thus leads to a clear and compelling picture of defect states in PE, thus clarifying the luminescence signatures of various defects in PE.
II. MODELS AND METHODS
Perfect crystalline PE consists of chains of singly bonded —CH2— groups, of which all the carbon atoms display sp3 hybridization. In this work, 1 × 2 × 3 supercells of crystalline PE, containing of 24 —CH2— groups, are constructed with and without defects as the models for our calculations. Six kinds of defects, illustrated in Fig. 1, are typically considered to exist in PE chains. A carbonyl defect (C=O) is created when the two H atoms of a —CH2— group are replaced by an O atom doubly bonded to the central C atom, which now displays sp2 hybridization. If the two H atoms of —CH2— are replaced by a=CH2 group, we have a vinyl defect. When each of two adjacent C atoms loses one H atom, a double bond (—CH=CH—) may be formed between them. The combination of a double bond and a carbonyl is dienone (—CH=CH—C=O). In the case of a hydroxyl defect, a H atom is replaced by a —OH group. Along the PE chains, an alternating pattern of single and double bonds leads to conjugated double bonds (—CH=CH—CH=CH—).
B. DFT computations
Our DFT calculations were performed with the Vienna ab-initio simulation package (VASP).25–27 The (folded) Brillouin zone of the supercell is sampled by a 2 × 2 × 2 k-point Monkhorst-Pack mesh28 while plane waves with kinetic energy up to 400 eV were included in the basis set. van der Waals interactions are considered within the scheme developed by Tkatchenko and Scheffler.29 Relaxed geometries were obtained with Perdew-Burke-Ernzerhof (PBE) XC functional,30 which were then used to compute the thermodynamic and optical transition levels with the PBE and the Heyd-Scuseria-Ernzerhof (HSE)31–33 XC functionals, and many-body perturbation theory GW method34–36 [including the single-shot GW (G0W0) and the fully self-consistent GW (scGW) with vertex corrections]. In the case of the HSE XC functional, we used the HSE06 version with the mixing coefficient α = 0.25 and the screening parameter ω = 0.2 Å−1. The band gap Eg of perfect PE calculated with PBE, HSE06, G0W0, and scGW methods are 7.00 eV, 8.39 eV, 9.02 eV, and 8.95 eV, respectively, with the scGW result being closest to the experimental value (8.80 eV).37 Because the accuracy of the computed emission energies involving the charge recombination process is affected by the accuracy of electronic structure calculations of perfect PE, the scGW method (hereafter referred as GW), rather than G0W0, is used in the subsequent calculations.
C. Optical transitions in PE
1. Intra-defect state transitions
By absorbing an amount of energy (Ea), a defect in PE may be transformed from the singlet ground state S0 into either the lowest-energy excited singlet state S1 or a higher-energy excited singlet state, e.g., S2 — see Fig. 2(a). In the latter case, processes like vibrational relaxation or internal conversion often follow, driving the defect to S1. The subsequent S1 → S0 transition may either be direct, i.e., fluorescence, or indirect, going through the triplet state T1 state by intersystem crossing, i.e., phosphorescence. The S0, S1, S2, and T1 states of double bond, conjugated double bond, vinyl, and hydroxyl defects are (π)2(π∗)0, (π)+1(π∗)−1, (π)+1(π∗)0(σ∗)−1, and (π)+1(π∗)+1, respectively, where the superscripts (0, 1, and 2) denote the number of electrons occupied in the corresponding state while ± represents the up/down electron spin. For carbonyl, S1 and S2, respectively, correspond to (n)+1(π∗)−1 and (n)+1(π∗)0(σ∗)−1, while T1 is (n)+1(π∗)+1 due to the existence of n-nonbonding orbital. However, because the carbonyl group is conjugated with a double bond in the dienone defect, the excited states of dienone are (π)+1(n)2(π∗)−1 for S1, (π)+1(n)2(π∗)0(σ∗)−1 for S2, and (π)+1(n)2(π∗)+1 for T1.38 The emission energies Ee of either the S1 → S0 or the T1 → S0 transition were calculated as Ee = Eexcited − Eground∗, where Eexcited and Eground∗ are the spin polarized DFT total energy of the fully relaxed defect at its excited state (S1 or T1) and the DFT energy of the defect at the ground state (S0) using the atomic configuration of the excited state, respectively.39 The relaxation energy of the defect at ground state may contribute to the broad PL peaks; thus, the zero-phonon line (ZPL), given by EZPL = Eexcited − Eground, is also calculated and used to compare with the experimental values. Here, Eground corresponds to the DFT energy of the fully relaxed defect at ground state.
2. Charge recombination processes
Photon emission can also be associated with charge recombination as in EL and TL. The scenario, as sketched in Fig. 2(b), is rather complicated with the involvement of the defect energy levels. The free charges in the conduction band, which were pumped from either the valence band by x-ray or γ-irradiation radiation in TL or the electrodes in EL, may be trapped by (shallow or deep) defect levels. While the trapped electrons can recombine directly with holes in the valence band (① process), those from deep traps can also be transferred to the excited states of the recombination centers by direct tunneling before relaxing to the lowest-excited level for the final recombination (② process). Electrons from shallow traps, on the other hand, can move to the excited states of the recombination centers by going through the conduction band (③ process). In reality, the electronic transitions between different energy states (i.e., defect levels, CBM, and VBM) can occur in two ways, differing by the time scale of the measurement process. Thermodynamic transitions (involving long time scales) will evolve the initial to the final charge state at its respective equilibrium geometries. Optical transitions will occur at much shorter time scales, bringing the system to the final charge state while retaining the equilibrium geometry of the initial charge state (see Fig. 2(c)). Clearly, determining the defect level positions accurately is the key to evaluate the emission energies from the charge recombination processes.
Similar to the case of intra-defect transitions, a formally correct approach to determine the defect levels and the associated transition energies is via total energy differences of PE with defects at initial and final charge states.13,14,40 The local equilibrium atomic configurations are different for the defects in different charge states,13 as illustrated in Fig. 2(c). A configuration–coordinate diagram of a defect in two charge states, 0 (q) and −1 (q′), is shown in Fig. 2(c), in which the minima of the potential energy surfaces represent the ground state of the defect in charge state q = 0 and q′ = − 1. In the case of thermodynamic transition, the defect has enough time to relax from its initial ground state at q = 0 into its new ground states at q′ = − 1 or vice versa, as shown using red double arrowhead line in Fig. 2(c). In order to get this charge transition level, the formation energies as a function of Fermi energy (EF) for this defect at charge states q = 0 and q′ = − 1 are computed and shown in Fig. 2(d), where Rq represents the equilibrium structures in charge state q. Based on Fig. 2(c), the thermodynamic transition level μ(0/ − 1) corresponds to the crossover point between and (solid lines in Fig. 2(d)).
In general, the thermodynamic transition levels μ(q/q′) is the Fermi energy at which defects in two different charge states q and q′ are at thermodynamic equilibrium and are given by13,14,40
Here, is the formation energy of the q-charged defect at its equilibrium structure Rq, which can be obtained from DFT calculations. The Fermi energy is taken from VBM to CBM of defect-free PE. In this work, all energies are referenced to the averaged electrostatic potential of PE far away from the defect location. The electrostatic interactions of charged defects due to the periodicity and the finite supercell sizes are corrected by first-order monopole corrections in all defect cases considered. The correct positions of the defect levels with respect to the VBM and CBM are needed to explain the transitions between these energy levels. Because DFT at the PBE and HSE06 levels does not offer such information due to the incorrect position of VBM (and of course, CBM), and as the GW method does not provide total energy, we computed the transition level with PBE and HSE06 and then amended the obtained positions by the VBM shifts (ΔEV BM = V BMGW − V BMHSE06) when the GW method is used, expressed as
The subscripts (HSE06 and GW) are refer to the method used to calculate the relevant quantities.
In the case of optical transition, the atomic configuration of the defect at the initial charge state q is fixed even though charge transition exists, i.e., it is a vertical transition which is shown using a blue arrowhead line in Fig. 2(c). Because the optical transition depends on the direction of charge transfer, two kinds of optical energies are possible, as shown in Fig. 2(c): emission energy (Ee) for charge transfer from initial state 0 to final state −1 and absorption energy (Ea) for the reverse process. The method to determine the optical transition level is similar to the previous case, but the energy of the final state is computed using the equilibrium structure of the initial state, such as and in Fig. 2(d). Therefore, the optical transition level from initial state q to final state q′ is defined as
where is the defect formation energy in the charge state q′ corresponding to the equilibrium structure Rq of the initial charge state q. The corresponding GW optical transition level is then expressed as
III. RESULTS AND DISCUSSIONS
A. Intra-defect state transition
To explain the physical origin of the observed PL-phosphorescence peaks in PE, the total energy differences of various neutral defects at their ground and excited states were calculated with the PBE and HSE06 functionals. The emission energies and the ZPL of triplet to ground state transition (T1 → S0) for all defects are shown in Table I, together with experimental PL results. The dienone case alone presented convergence issues with HSE06 functional, and hence, this result is not presented in Table I. However, given the reasonable correlation between the PBE and HSE06 results, the PBE results for this defect may be viewed as acceptable. For ease of visualization, the PBE results of the defect at excited and ground state are also shown in Fig. 3. The experimental PL levels in Fig. 3 are determined by the energy differences between the experimental emission energy and the S0 ground state.
|.||Ee .||EZPL .||.|
|Defects .||PBE .||HSE06 .||PBE .||HSE06 .||Experiments .|
|Conjugated double bond||1.71||1.51||2.17||2.09||2.90b|
|.||Ee .||EZPL .||.|
|Defects .||PBE .||HSE06 .||PBE .||HSE06 .||Experiments .|
|Conjugated double bond||1.71||1.51||2.17||2.09||2.90b|
The consistency between the computed and experimental emission energies of carbonyl and dienone defects indicates that the PL emission bands of dienone and carbonyl involve the transition from the (π)+1(π∗)+1 excited state to the (π)2(π∗)0 ground state, and the transition from the (n)+1(π∗)−1 excited state to the (n)2(π∗)0 ground state, respectively. A small discrepancy between the computed and the measured results for conjugated double bond defect still remains. Possible reasons for this disagreement might include some uncertainties in the experimental data, or more likely, because in the present calculations, the defects are modeled in the crystalline PE environment. Nevertheless, we can conclude that the luminescence data for conjugated double bond defect are related to the transition from the (π)+1(π∗)+1 excited state to the (π)2(π∗)0 ground state. For hydroxyl, double bond, and vinyl defects, the low vertical transition energies, i.e., Ee, indicate that the relaxation from triplet to ground states may either be radiationless or hard to detect.
B. Charge recombination processes
1. Defect geometries
The relaxed structures of PE with defects at different charge states q [denoted by (defect)q, q = 0, −1, and +1] are shown in Fig. 4. The variation of geometry parameters (bond angle ∠ and bond length l) of (defect)q indeed shows that the equilibrium structures corresponding to different charge states are different. For example, the C=O bond length of carbonyl increases from 1.228 Å for the neutral state to 1.315 Å for the −1 charge state, because the extra electron is localized at the C=O bond and leads to bond weakening. Whereas, compared to the (carbonyl)0, the decrease of the C=O bond length in (carbonyl)+1 is due to the loss of an electron. As another example, we note that a rotation of about 90° can be observed for C=C part of (double bond)−1 and (double bond)+1 compared to the neutral case. This is because both positive and negative charges can weaken the C=C bond and make the PE chain free to rotate. All these relaxations for (defect)q make contribution to the broad luminescence peaks in experiments. Therefore, in this work, all levels ranging from , which is relevant to the emission energy, to μ(q/q′) are used to compare with experimental results.
2. Charge transition levels
In order to characterize the defect levels in PE, which are responsible for the observed luminescence signatures (especially the EL and TL through charging/discharging of the defect levels as per the mechanisms outlined in Fig. 2(b)), the thermodynamic and optical transition levels for the six typical defects in PE have been calculated. The results obtained from various methods (solid lines) and available EL and TL experimental data are shown in Fig. 3. As can be seen, the (0/+1) charge transition levels for carbonyl, dienone, conjugated double bond, and double bond are close to the VBM, indicating that these four defects are easy to lose an electron and become positive charge state. The associated thermodynamic (i.e., μ(0/ − 1) and μ(0/ + 1)) and optical (i.e., , , , and ) transition levels are represented with blue, wine, red, magenta, violet, and orange solid lines, respectively. Because of the broad peak observed in experiments, the levels from to μ(0/ − 1) or from to μ(0/ + 1) are used to compare with experimental levels, as shown by the shaded region in Fig. 3.
Given the computed defect level placements, excited state energies, and the measured luminescence signatures portrayed in Fig. 3 in a unified manner, we propose that the mechanism ② in Fig. 2(b) underlies the EL spectra of PE containing carbonyl, dienone, hydroxyl, conjugated double bond, and double bond defects, and that mechanism ① underlies the measured TL spectra of PE containing vinyl defects. More specifically, in the former cases (when we believe mechanism ② is in operation), an electron from the of such defects may first tunnel to their (energetically well-positioned) S2 excited state, then pass to the triplet (T1) excited state by internal conversion and intersystem crossing, and finally transfer to the level, accompanied by with emission of photons, i.e., (defect)−1 + (defect)+1 → (defect)0. The entire process can be represented by (0/ − 1) → S2 → S1 → T1 → (0/ + 1), with the last step leading to the observed EL signature. On the other hand, in the case of the vinyl defect, both and are close to the experimental result. This implies that the transitions of electrons from the conduction band to the of vinyl and electron transferred from to the valence band are both likely the origin of the luminescence caused by the vinyl defect. In other words, we use the computed defect charge transition level and excited state placements to arrive at the simplest theory that matches with observations. The proposed recombination pathways, the computed transition/emission energies (at various levels of theory), and available measured EL and TL signatures are listed in Table II.
|Defects .||Path .||PBE .||HSE06 .||GW .||Experimental values .|
|Conjugated double bond||2.34–2.90||1.76–2.51||1.76–2.51||…|
|Defects .||Path .||PBE .||HSE06 .||GW .||Experimental values .|
|Conjugated double bond||2.34–2.90||1.76–2.51||1.76–2.51||…|
It is also worth noting a few curious details related to the computed results. When referenced to the average electrostatic potential, the charge transition levels calculated with PBE agree well with those based on the HSE06, and the GW method, as shown in Fig. 3. Unlike the charge transition levels, a remarkable variation (with the level of theory used) of the band edges of PE is observed in Fig. 3, which may impact the comparison with experimental emission energies. To show the role of band edges, the emission energies of defects computed with the PBE, HSE06, and the GW method are listed in Table II, together with available experimental results. For carbonyl, dienone, hydroxyl, conjugated double bond, and double bond defects, the emission energy ranges from to μ(0/ − 1) − μ(0/ + 1) − Er, where Er is the energy differences of the fully relaxed defect at S2 and T1 excited states, calculated with spin polarized PBE and HSE06 functionals.39 In the case of vinyl, the emission energy ranges from to CBM − μ(0/ + 1), or from to μ(0/ − 1) − V BM. We note that the computed emission energies of carbonyl, dienone, and vinyl with all functionals agree well with experimental values. This indicates that both charge transition levels and emission energies are captured well by the PBE functional, if a suitable reference energy is used.
3. Kohn-Sham levels
To complete the discussion, and to put our work in the context of the existing literature pertaining to defects and defect states in PE, we also include the one-electron Kohn-Sham defect levels for various defects in PE in Fig. 3. These are shown using black (unoccupied states) and red (occupied states) dashed lines in Fig. 3. While the Kohn-Sham defect levels are in reasonable and qualitative agreement with the charge transition levels, some important observations should be made. First, even when the PBE and HSE06 average electrostatic potentials are aligned, the one-electron levels do not line up (unlike the charge transition levels). This indicates that the one-electron levels are strongly dependent on the level of theory used. Second, the emission energies that one would arrive at purely using the one-electron energies and the band edges of the same theoretical treatment would be in stark disagreement with experiments. Thus, interpreting defect spectroscopic signatures using Kohn-Sham levels should be performed with cautions.
In summary, we have presented a detailed first-principles computational study of chemical defects in PE and have quantitatively determined the emission and absorption energies derived from intra-defect state transition and charge recombination processes. For the intra-defect state transition, the emission energies are calculated by total energy differences of the neutral defect at excited and ground states. These results are in good agreement with the measured photoluminescence bands of PE. To assign the electroluminescence and thermoluminescence peaks involving the charge recombination process, we go beyond traditional density functional theory calculations within the one-electron framework. In particular, we determine the defect levels in terms of thermodynamic and optical charge transition levels via total energy calculations of neutral and charged defects. Consequently, by combining the defect level positions and the energy differences of defects at higher and lowest excited states, the most likely emission mechanisms and emission energies have been determined, leading to excellent correspondence with available experimental results. Finally, we suggest that this computational scheme may be used to interpret the luminescence data of other organic polymers as well.
This work is supported by a Multidisciplinary University Research Initiative (MURI) grant from the Office of Naval Research. Discussions with Christian Laurent and Gilbert Teyssedre are gratefully acknowledged. The authors also thank one of the anonymous reviewers, whose diligent comments have lead to significant improvements in the quality, reach, and impact of the manuscript.