Models of bipolar charge transport in polyethylene

We introduce and develop two bipolar transport models which are based on appreciably different physical assumptions regarding the distribution function in the energy levels of trap states. In the first model, conduction is described by an effective mobility of the carriers and the accumulation of stored space charge is taken into account through a single trapping level. In the second model the hypothesis of an exponential distribution function of trap depth is made, with conduction taking place via a hopping process from site to site. The results of simulations of the two models are compared with experimental data for the external current and the space-time evolution of the electrical space charge distribution. The two descriptions are evaluated in a critical way, and the prospects for these models to adequately describe real systems are given. © 2006 American Institute of Physics. DOI: 10.1063/1.2375010


I. INTRODUCTION
The use of polymeric materials, especially polyethylene as the insulation for high voltage ͑HV͒ cables, has grown steadily since they were introduced more than 30 years ago.Today polyethylene is rapidly becoming the preferred insulation material for even the highest transmission voltages.In respect to traditional oil-impregnated paper insulation, polyethylene exhibits quite beneficial characteristics, these being high reliability, low dielectric losses, and low environmental impact.However, under certain conditions, polyethylene may suffer from chemical, physical, and electrical aging or degradation after long-standing operation under voltage. 1 The degradation of the polyethylene may lead to electrical breakdown in the insulation of an alternating current ͑ac͒ or direct current ͑dc͒ cable.All of these factors when combined over time make the prediction of the probable working life of extruded cable very difficult.A problem for applications in high voltage dc ͑HVDC͒ transport is the propensity for polymeric insulation to accumulate charge through either injection at the interfaces or internal generation processes.These so-called space charges may produce field distortion within the insulation, and also catastrophic damage during the transmission of electrical energy.An understanding of space charge accumulation and its link to dielectric breakdown would be a major scientific achievement as well as facilitating the prediction of cable lifetime and the design of better cables.Nevertheless, the space charge accumulation and the related trapping-detrapping phenomena are still poorly understood.Over recent decades, a great deal of effort has been expended in developing experimental tools for space charge characterization, [2][3][4][5][6] in measuring space charges profiles, [7][8][9] or in identifying the different kinds of charges. 10,11][14][15][16] We have initiated an activity which aims at modeling charge transport phenomena in a synthetic insulator largely used in the high voltage area: polyethylene.The aims are to account for the internal distribution of field as a function of stress, which could be used in the future to feed aging models, since, whatever the aging scenario considered, the local field is an important parameter.The transport model must be able to account for the distribution of electric charges ͑and of electric field͒ in nonstationary situations.The approach must be general enough to be applicable to a wide variety of insulators.
The aim of the present contribution is to compare two bipolar transport models that we have developed recently.The two models have been designed with the same objective, being to simulate charge transport in nonstationary conditions, but they are based on appreciably different physical assumptions regarding the distribution function in trap levels.For both models, carriers are provided only by charge injection at the electrodes.In the first model, conduction is described by an effective mobility of carriers, whereas for charge storage processes, space charge is taken into account through trapping in deep traps at a single specific energy level.In the second model, one makes the hypothesis of an exponential distribution function of trap depth, with conduction occurring via a thermally driven hopping process from site to site.The results produced by simulation of the two models are compared with experimental data for the external current and the space-time evolution of the electric charge distribution.

A. Common features
Even when neglecting dipolar processes and diffusion one generally has to solve the following coupled equations considering a one-dimensional ͑1D͒ problem along the spatial coordinate x, whatever the model used to describe charge transport: for transport equation, where j is the transport current associated with each kind of carrier of density n and charge q, is the mobility, D is the diffusion coefficient, E is the applied field, and is the net density of charge.Equations ͑1͒ and ͑2͒ are written for each kind of carrier defined in the model.The term s is the source term, i.e., it encompasses changes in local density due to processes other than transport, such as the internal generation of charges, recombination, etc.These equations have a specific form for the interfaces and are complemented by boundary conditions ͑e.g., applied voltage, etc.͒.
In the models, we have supposed that charge generation results from injection at the electrodes according to a Schottky law, being for electrons and holes, where j e ͑0,t͒ and j h ͑D , t͒ are the fluxes of electrons and holes at cathode and anode, respectively, T is the temperature, A = 1.2ϫ 10 6 Am −2 K −2 is the Richardson constant, D is the interelectrode spacing, and w ei and w hi are the injection barriers for electrons and holes.
There is no extraction barrier, the extraction fluxes being j e ͑D , t͒ and j h ͑0,t͒ for electrons and holes at anode and cathode, respectively; j e,h ͑z,t͒ = e,h E͑z,t͒ f͑e,h͒ ͑z,t͒.͑6͒ The total current density, J͑t͒ is obtained from second Maxwell's equation, that is The first term of the right-hand side is the conduction contribution and the second term is the displacement current.It can be shown that Eq. ͑7͒ can be rewritten as Note that dV / dt is null in the present simulation conditions.

B. Model I
Model I, which has already been described in several papers, 12,17,18 is depicted in Fig. 1.Two kinds of carriers are considered, being either trapped or mobile.A mobile electron in the conduction band ͑hole in the valence band͒ is associated with an effective mobility.This mobility accounts for the possible trapping and detrapping in shallow traps, in which the time of residence is short ͑Ϸ10 −12 s͒. 12,19,20Deep trapping ͑coefficients B i ͒ is described through a single trapping level for each kind of carrier.Charge carriers have a certain probability to escape from deep traps by overcoming a potential barrier that is included in the detrapping coefficients D i .The recombination of carriers is accounted for considering different coefficients S i for the several electron-hole pairs.After recombination, the trap is released and trapping, thus becomes possible again.A small quantity of mobile negative and positive charges ͑up to 0.5 C m −3 ͒ is supposed to be initially distributed uniformly within the dielectric while keeping it electrically neutral ͑the local net charge density is null͒.Taking into account the nonpolar character of polyethylene, dipolar polarization has been neglected.Also, the internal generation of carriers is not considered.An example of an expression for the source term of Eq. ͑3͒ is given for mobile electrons in Eq. ͑9͒,

͑9͒
where S i are the recombination coefficients, B e is the trapping coefficient for electrons, n e and n h are mobile electron and hole densities, n et and n ht are the densities of trapped electrons and holes, and n 0et and n 0ht are the trap densities for electrons and holes.Note that the source term is expressed here in terms of the carrier density whereas it is in charge density in earlier definitions. 12,17The detrapping probability is defined by a detrapping coefficient, for each species, which is of the form where is the attempt to escape frequency, which has been set to k B T / h = 6.2ϫ 10 12 s −1 at room temperature, T is the temperature, and w is the detrapping barrier.

C. Model II
The main feature of model II is that an exponential distribution of trap depths is considered and that the same ensemble of carriers moves site to site and participates in deep trapping.For the sake of simplification, charge recombination has not been included in the model.With these two hypotheses, the source term in Eq. ͑3͒ is null in this case.

Trap distribution
The exponential distribution ͑Fig.2͒ has been truncated by defining a maximum trap depth in order to avoid extremely low values of mobility at low charge density, Here N t͑e͒ and N t͑h͒ ͑in m −3 J −1 ͒ are the trap density distributions, respectively, for electrons and holes, characterized by a shape parameter T 0 , the preexponential factor NЈ, and maximum trap depth ⌬ m .

Transport and trapping
Traps are supposed to be filled from the deepest levels upwards. 13This means that an upper filled level, ⌬ f ͑x , t͒, can be defined, which is time and space dependent.The relation between carrier density, for one specie, and ⌬ f is of the form which gives Charge transport within insulating polymers is often described by a hopping mechanism in which carriers move from site to site by getting over a potential barrier.In our case the hopping charges essentially come from the highest filled trap state at a depth ⌬ f , and the resulting mobility has the form 13

͑E͒
where d is the average distance between traps and e is the elementary charge.
When T Ӷ T 0 , only a fraction n f of the total carrier density n is involved in transport: this corresponds effectively to carriers trapped at an energy within k B T in respect to ⌬ f , 13,21,22 The average intersite distance d is related to the trap density according to

͑16͒
Hence, once the shape parameters for the distribution function have been settled, the mobility is entirely defined regarding its temperature, field, and carrier density dependencies.

Model parameters
One way to proceed for parameterizing the distribution is to consider literature data on trap density estimates. 23uirke and co-workers 19,24,25 developed molecular simulation aimed at estimating the trap depth associated with given chemicals in polyethylene, along with those for physical traps associated with disorder in chain conformation.The results of these simulations for electronic carriers are shown in Fig. 3.In order to compare the continuous distribution used in model II and the discrete distribution obtained by molecular simulation, we have integrated the continuous distribution function on adjacent intervals of width of 0.2 eV between 0 and 1.2 eV, setting the distribution parameters to NЈ =10 47 m −3 J −1 and T 0 = 1160 K.The exponential distribution reasonably accounts for these molecular simulation results.Actually, simulation data are available only for electron traps, and only a limited amount of chemicals has been considered in this work.In addition, these simulations were relevant to cross-linked polyethylene rather than low density polyethylene ͑LDPE͒.Nevertheless, this comparison shows that the hypothesis of an exponential distribution of traps can be realistic.

D. Resolution and application
The numeric techniques implemented to solve the set of equations are basically the same for the two models. 26riefly, the electric field and potential in the dielectric are calculated with a boundary element method ͑BEM͒. 27With this method, the computation is more rigorous than by using discretization of Poisson's equation, and furthermore the field is known on every point in the dielectric, including the electrodes.Our numerical model employs a specific numerical method of solution of the continuity equation in order to accurately resolve the steep fronts that may appear in the electron and hole densities.This numerical method is based on a numerical scheme initially developed by Leonard 28 ͑quickest scheme͒ that we extended to very general situations.This scheme is employed with a specific flux limiter that guarantees that all particle densities are positive.The cell used for computation is divided into elements ⌬x, of varying length, being tightened close to the electrodes in order to optimize numeric computation.The time step in the computation is chosen automatically depending on the variation rate in the computed parameters.It must satisfy the condition of Courant-Friedrichs-Lewy, 29 ͑CFL͒ in a manner so as to guarantee the stability of the numerical scheme.
In respect to model I, model II may appear simpler, since only two ensembles of parameters are defined: that associated with carrier generation at the electrodes, being the same as for model I, and the set of quantities defining the shape of the trap-energy distribution.Since mobile and trapped species are not differentiated, there are only two kinds of species ͑instead of four͒.In addition, the source terms are null.Table I compares the main features of the two models.
Model I has been applied successfully to simulate the time dependence of space charge density profiles, external current, and electroluminescence considering measurements realized in charging/discharging conditions, along with various protocols for dc stress applications. 18,30Such work led to a set of parameters, reported in Table II that appears suitable to describe the behavior of LDPE.The first objective of this work is to compare the outputs of the two models for initial conditions that are as close as possible.In Sec.IV, the parametrization of model II is improved in respect to the experimental behavior.

III. COMPARISON OF MODEL OUPUTS
In this section, we attempt to parametrize model II as closely as possible to model I and we discuss the differences between the two approaches.As an experimental basis, we have considered the charging and discharging behaviors of a 150 m thick LDPE film with 40 kV/ mm applied field and 3 h of charging/discharging time at room temperature.Ex-   perimental data on space charge measurements 18 have been obtained by the pulsed electroacoustic ͑PEA͒ technique.
The results produced by either model cannot be directly compared to experimental data, especially the space charge profiles obtained by the PEA method, firstly because only internal charges are produced by the simulation that do not influence charges on the electrodes; secondly, profiles obtained by the PEA method are, as with any other technique, accompanied by uncertainty in the position of the charges.As it is not possible in practice to correct experimental data for the uncertainty associated with charge position, we have applied a numerical filter to our data in order to lose part of the resolution reached by the simulated profiles. 31The figures produced for the space charge evolution are maps of the net charge density inside the sample as a function of time.Each map has three dimensions: thickness ͑Y axis͒, time ͑X axis͒, and charge density, which is represented by a color scale.Space charge profiles are produced at time intervals that are shorter at the beginning of the polarization and depolarization phases in order to improve resolution.Hence, the time scale is not linear, but is the same for experiment and simulation.
We have considered that charges can be present initially in the material: the density for positive and negative charges is the same so that the net charge density, i.e., that probed by any space charge measurement method, is null.This hypothesis did lead to consistency between experimental and simulated electroluminescence transients at short time when using model I. 30

A. Mobility versus field and charge density for model II
Whereas injection conditions could be taken as rigorously the same for the two models, the hypotheses used did not allow the same to be said of the mobility.A fixed mobility is set in model I whereas Eq. ͑14͒ shows that is dependent on charge density and field, and that the three parameters of the distribution control this dependency,

͑17͒
Even when n = 0 and E = 0, the mobility is dependent on the three parameters and several combinations of NЈ, ⌬ m , and T 0 may give a preset mobility.
When kT 0 ӷ⌬ m , which is so for T 0 of the order of 10 3 K and ⌬ m of the order of 1 eV, the intertrap distance ͓Eq.͑16͔͒ can be approximated to d = ͑N Ј k B T 0 ͒ −1/3 so that at low car- rier density, the parameters controlling ͑n , E͒ are NЈ, T 0 , and ⌬ m .However, it can be readily observed that the variation of mobility with carrier density is stronger when T 0 is small.Therefore an approach to the conditions of a mobility independent from the charge density requires a relatively high T 0 .The variation of mobility with field will be weaker as d becomes smaller, i.e., NЈ, T 0 large to a first approxima-tion.There is finally a constraint on the value of mobility: we have arbitrarily chosen a value of ⌬ m such that ͑0,0͒ equals that used in model I.
Table II shows the combinations of parameters that have been tested ͑sets II.1 and II.2͒, and Fig. 4 shows how the mobility varies with field for both electrons and holes in these conditions.The mobility significantly increases at fields in excess of 50 kV/ mm.It is independent of charge density up to at least 10 3 C/m 3 .

B. Results
Figure 5 compares space charge patterns obtained using the two models for a 150 m thick sample under 40 kV/ mm dc stress at 293 K.The charging and discharging times were 3 h.In both cases, space charge profiles appear dominated by positive carriers, a feature which arises from the significantly lower Schottky barrier for holes ͑1.16 eV͒ as compared to electrons ͑1.27 eV͒.It can readily be seen that different results are obtained in the two models.The accumulated charge density appears lower in the case of model II.Charge release during depolarization can be more clearly seen in the case of model I.The transit time for the first positive charge front also changes, depending on the situation.It is about 20 s for set I.1, 1000 s for II.1, and beyond 3 h for II.2.
Figure 6 compares the charging current transients obtained in the two cases.Here again, quite different behaviors are predicted.Model I predicts a continuous decrease of the current, whereas in model II the current increases, forms a peak, decreases for a while, and then increases again.In model I, the current increased up to a maximum and then decreased when supposing that no initial charges were present. 12Such shapes, especially the peak in current transient, are more generally characteristic of space-chargelimited current transients. 32In our simulations, the initial current is higher when the initial charge density is large.Obtaining a decaying transient current is a matter of balance between charge injection and transport, a favorable situation being roughly strong injection associated with slow transit.

C. Discussion
Supposing that positive carriers are dominant and that there is not much field distortion within the insulation, the transit time for carriers being at the charge front should be governed by the group velocity of carriers defined by where n h and n h are the density of mobile holes and the total density of holes, respectively, and E a is the applied field.In the case of model II, the fraction of mobile carriers is readily obtained from Eq. ͑15͒.The expected transit times for sets II.expected comes mainly from field distortion due to space charge ͑the field at the charge front exceeds the applied field by up to 20% of the applied field͒, and possibly to some extent to the value of mobility which is slightly above 2 ϫ 10 −13 m 2 V −1 s −1 at 40 kV/ mm.In the case of set II.2, it can be observed that the velocity of charges tends to increase once they have crossed about half of the insulation, and this behavior results from field distortion ͑up to 20% intensifica-tion͒.Hence, for the case II.1, positive charges move more freely in the insulation and less accumulation than in case II.2 is observed, since accumulation is a matter of balance between generation and transport.This mechanism is confirmed considering Fig. 6 where the external current is higher in case II.1.For model I, the transit time for the injected charge front to reach the counterelectrode is much shorter, being about 20 s, even if this is not clearly evidenced in the figures, as commented in the following.In addition to transport, trapping, and detrapping, model I allows for recombination processes.The criticality study achieved on this aspect 33 showed that for recombination parameters in the range of 10 −1 -10 −5 m 3 C −1 s −1 , recombination processes contribute only marginally to the change in local density of carriers.Hence, recombination processes do not constitute the leading factor in the difference between the two models.Neither does the assumption of an initial density of charges since the space charge and current profiles are not influenced at times beyond 10 s, for h = e ͑t =0͒ in the range of 0 -0.5 C / m 3 .As mobility is quasiconstant in either case, the differences between the two models in the charging behavior have to be found in the number of carriers available for conduction.In the case of model II, the fraction of charges involved in conduction ͓Eq.͑15͔͒ is of the orders of 1.5ϫ 10 −2 and 10 −3 for the set of parameters II.1 and II.2, respectively.For model I, the fraction of mobile positive charge with respect to the total amount of positive charge carriers is more difficult to estimate since it results from trapping, detrapping, and recombination probabilities and it is dependent on time and space.However, a simple analytical solution for the transit time of the charge front injected at time zero can be obtained under the approximations that internal field distortion due to space charge and trapped charge densities is negligible: only trapping is allowed for during transport.With these hypotheses, the transport equation for mobile holes can be rewritten as

͑19͒
Considering that n h ͑x , t͒ = 0 for t ഛ 0, the solution is of the form Equation ͑20͒ shows that the injected front is attenuated during propagation and that no qualitative distortion of the charge front occurs during transport.From the parameter set I.1, the time necessary for the front to reach the counterelectrode ͑x = wt = 150 m͒ is estimated to be 18.8 s, in good agreement with the simulation results.A strong attenuation of the mobile hole density is predicted ͑about 2.5% of the charges injected at time 0 reach the counterelectrode͒: this feature explains why in simulated profiles of Fig. 5͑a͒ a time much longer than 20 s is necessary for observing an appreciable amount of positive carriers approaching the cathode.Even though there is an attenuation of mobile carrier density due to trapping, the fraction of mobile carriers remains much higher than for model II; as a consequence, the charging current is stronger in model I at short time.Model I gives stronger charge accumulation and less current density than model II in the long term.The only explanation for this behavior is that as time goes on, the fraction of carriers involved in conduction becomes smaller, i.e., the proportion of trapped charges is continuously increasing.Carriers are assumed to be mobile immediately after being injected.They are progressively trapped during transport.As time goes on, fewer charges are injected due to the field decrease at the anode, and hence the proportion of trapped charges grows.Of course, other phenomena such as detrapping and recombination contribute to the balance between trapped and mobile carriers.
It can be seen in Fig 5͑b͒ ͑model II͒ that the positive charge density decreases in the vicinity of the cathode at long times.This corresponds to injection of negative carriers.Figure 5͑d͒ shows density profiles for the set of parameters II.1 using a different representation: post-treatment of data was not achieved and, in addition, the charge density color scale is different.The injection of negative carriers clearly appears in this picture.The increase in the charging current observed at long time scale ͑Fig.6͒ corresponds to the response of the material when electron transport significantly contributes to the external current.Injected negative carriers progressively penetrate into the insulation, thus diminishing the net charge.Model I also predicts negative charge injection and drift in the bulk.Negative charges are the dominant carriers up to 20 m from the cathode at the end of the polarization stage.

A. Distribution parameters
In the above attempt to compare the two models for a near identical parameter system, it appears that the predicted external transient current for model II does not have the same shape as the experimental behavior ͑Fig.6͒.Also, the internal charge density appears less than the experimental one ͓the latter is shown in Fig. 7͑a͔͒.Consequently a new set of parameters has been looked for in order to find an agreement with the experimental results.
As stated previously, in situations dominated by transport of one kind of carrier, the models predict that a decreasing charging current is obtained once the front of charges has reached the counterelectrode.However, the experimental current is continuously decreasing in the time range when data are available, whereas a front of positive carriers moves slowly in the space charge measurements.There could be other origins to the current at short time such as orientation polarization.Polyethylene is a weakly polar material, and though this possibility cannot be completely discarded due to a lack of experimental data on the order of magnitude of polarization effects, we suppose for the moment that the response is dominated by transport.Some simulation conditions break the above rule on the relation between charge front and transient current shape.First, if the response is dominated by the transport of preexisting carriers, then a decreasing charging current is obtained.Second, in situations where massive injection occurs associated with significant charge accumulation ͑strong field dis-tortion͒, again the current becomes continuously decreasing.A way to favor injection in respect to transport is to decrease the barrier height to injection and/or to decrease the fraction of mobile carriers by decreasing T 0 .The fit to the experimental behavior, especially as regards the transient current, has been improved by acting on these two factors.The final set of parameters is given in Table II, set II.3.As previously, the injection barrier for holes is significantly lower than that for electrons.
In the set of parameters II.3, T 0 is smaller than that ͑1160 K͒ estimated from the simulation data of Quirke and co-workers ͑cf.Fig. 3 for electron traps͒.In fact, the slope of the curve ͑in semilog coordinates͒ is governed mainly by the density of deep traps.As Quirke and co-workers have considered cross-linked polyethylene ͑XLPE͒ as a model material, they have incorporated cross-linking by-products as potential deep traps.Consequently, LDPE should contain a lower concentration of deep traps, and hence T 0 is expected to be smaller for LDPE.The preexponential factor NЈ e is about 10 45 m −3 J −1 , which is somewhat lower than that obtained by regression in Fig. 3 ͑10 46 -10 47 m −3 J −1 ͒, indicating that not necessarily all physical traps are involved in transport.Overall, the distribution parameters obtained can be considered as physically conceivable.

B. Results and discussion
With this set of parameters, the transient charging current ͑Fig.8͒ is predicted both quantitatively and qualitatively, FIG. 7. ͑Color online͒ Experimental and simulated net charge densities vs time using the optimized parameters of set II.3.t on and t off stand for the time of voltage application/short circuit.A and K stand for anode and cathode, respectively.Note that the color scale is not the same due to different softwares for handling data.The simulated charge density is overestimated in the charging period and the discharging is slower than expected.except at long charging times when possible transport of negative carriers comes into play.The charge density ͑Fig.7͒ is overestimated in the charging period.In the discharging period, charges apparently move too slowly in respect to the experimental behavior.This feature was already noticed in model I.In respect to case studies II.1 and II.2, mobility becomes strongly dependent on field and charge density, as shown in Fig. 9.For a charge density of the order of 30 C/m 3 , the mobility for E = 40 kV/ mm is of the same order as that of model I.However, for low charge densities ͑i.e., at short polarization time͒, the mobility is small ͑order of 10 −16 m 2 V −1 s −1 for holes͒, thereby favoring accumulation in respect to trap-to-trap hopping transport.
Experimentally, it can be estimated that the front of positive charges reaches the counterelectrode within roughly 2500 s.The transit time for the front of positive carriers is 4000 s in simulation, which is somewhat higher than the experimental value and which is consistent with that deduced from Eq. ͑18͒ ͑4900 s͒, taking into account velocity enhancement due to field distortion at the charge front.Again, the time at which the positive charge front reaches the counterelectrode corresponds to the peak at 4000 s in the transient current ͑Fig.8͒.It is worth pointing out here that simultaneous measurement of space charge and current at high field, i.e., in conditions where packetlike behavior was observed in XLPE, demonstrated that the external current is at a maximum at the time a packet reaches the counterelectrode. 34,35e anode field for set II.3 decreases very quickly from 300 s onwards, being of the order of 10 4 V / m.Hence, it is expected that the contribution from electron transport becomes significant at long times; it is, however, not necessarily so at 10 4 s since the negative charge extends up to only about 12 m from the cathode, with a density not exceeding −5 C/m 3 .
For low fields ͑as in the depolarization stage͒, the mobility is low.The minimum value for hole mobility is ͑0,0͒ =4ϫ 10 −17 m 2 V −1 s −1 .Discharging is observed anyway in space charge profiles since a relatively high charge density is present initially, and a large fraction ͑about 25%͒ of carriers is mobile.Figure 8͑b͒ shows discharging currents obtained experimentally after 10 5 s of polarization.For model II, though orders of magnitude of the current could be reproduced in a certain time interval, the depolarization current decays more slowly than experimentally measured.This is most probably related to an underestimated mobility during depolarization.In the case of model I.1, the depolarization current is even lower.As stated above, in this model the fraction of trapped charges is continuously growing during the polarization stage so that a large majority of carriers stay in deep traps after 3 h of polarization.Hence, in this case, the detrapping probability has to be considered rather than the mobility as the rate limiting step in depolarization.Clearly, depolarization currents could not be reproduced satisfactorily by either model.There could be two reasons for this.Firstly, depolarization features are primarily dependent on the actual space charge distribution set up at the end of the polarization stage.Experimental data were obtained in independent experiments, the polarization time being longer for current ͑10 5 s͒ than for space charge profiles ͑10 4 s͒, so that we actually do not know what was the space charge profile prior to the depolarization current measurements.Simultaneous measurement of current and space charge could be advantageous in this respect since it would provide the actual space charge distribution built up prior to discharging current measurement.Secondly, as stated previously, attempting to reconcile experimental and simulated transient currents could be misleading, as other processes, such as orientation polarization, and possibly ionic contributions could contribute to the current without observable effects in space charge measurements.

V. CONCLUSION
Two bipolar transport models aimed at accounting for the space charge accumulation and transient current in LDPE under dc stress have been presented and compared.The first model considers mobile and trapped carriers as distinct species, a field-independent effective mobility being associated with the former and a single deep trapping level to the latter.Exchanges between the different kinds of species are governed by trapping, detrapping, and recombination probabilities.In the second model, an exponential distribution of trap levels is considered, the same ensemble of carriers being involved in trapping and transport.In this case, a thermally assisted hopping mobility is considered, being charge density and field dependent.Carriers are provided by Schottky injec- tion in both cases.It is shown that the two models are not equivalent, regarding the kinetics of space charge buildup within the insulation and the transient currents.The proportion of trapped charges increases during the time under stress for model I, whereas it is constant in model II.As a consequence, model I predicts a more strongly decaying transient current and slower discharging.Features of the early calculations on transient space-charge-limited current, such as a peak in the external current when charges reach the counterelectrode, 22 are reproduced for model II.
For both models, though a fit to the experimental behavior could be obtained in the charging period, discharging was in general too slow.In this respect, we would like to stress the necessity to unravel the nature of charging and discharging currents: in the models, we have considered that current originates only from transport of injected carriers.Dipolar contributions have not been dealt with because polyethylene is normally a weakly polar material, and hence this contribution is expected to be low.However, low frequency dielectric spectroscopy data are not easily available in the literature to support this hypothesis and to state up to which current level this is actually so.In the same way, low mobility ionic species could contribute to the external current.This normally should give rise to heterocharge formation.However, the limited resolution of space charge measurement techniques close to the interface, possibly combined with injection ͑formation of homocharges͒, may prevent evidence of such processes being obtained from PEA data.Hence, attempting to explain space charge accumulation and transient currents by a single process may be misleading.

FIG. 1 .
FIG. 1. Schematic representation of the conduction and trapping coefficients for model I. S i are recombination coefficients, n e , n et , n ht , n h are mobile and trapped electron and hole densities, and B i and D i are trapping and de-trapping coefficients.
s 1 = ‫ץ‬n e ‫ץ‬t = − S 1 n ht n e − S 3 n h n e − B e n e ͩ1− n et n 0et ͪ+D e n et ,

FIG. 2 .
FIG. 2. Exponential distribution of trap levels for one kind of carriers.The upper filled level ⌬ f is variable as a function of time and space.

3 ͒ 5 FIG. 3 .
FIG. 3. Discrete distribution of electron trap density as a function of trapping energy.The circles are results of Quirke's molecular simulation of Meunier et al. ͑Ref.24͒.The continuous distribution function used in model II ͑solid line with NЈ =10 47 m −3 J −1 and T 0 = 1160 K͒ has been integrated on adjacent intervals of width of 0.2 eV ͑squares͒.

FIG. 4 .
FIG. 4. Electron and hole mobilities vs field for model II.Two sets of parameters are considered ͑II.1 and II.2͒.The carrier density is set to n e = n h = 0.The mobility significantly increases at field in excess of 50 kV/ mm.
FIG.5.͑Color online͒ Net charge density vs time for a 3 h / 3 h polarization/depolarization cycle simulated using parameters of TableII.t on and t off stand for the time of voltage application/short circuit.A and K stand for anode and cathode, respectively.The accumulated charge density is lower in the case of model II.The transit time for the positive charge front varies from ͑a͒ 20 s to ͑b͒ 1000 s ͑not clearly evidenced in the figures as commented in the discussion part͒ and to ͑c͒ 3 h.

FIG. 8 .
FIG. 8. Experimental charging and discharging current densities compared to results from models I and II ͑set II.3͒.The discharging currents correspond to previous polarization for 10 4 s for model I, for 10 4 and 10 5 s for model II, and for 10 5 s in experiment.Whereas the charging current can be approximately fitted with either model, the discharging current decays more slowly than experimentally measured.

FIG. 9 .
FIG. 9. Mobility as a function of charge density ͑for E =40 kV/mm͒ and field ͑for =0͒ for parameter set II.3.The mobility is significantly dependent on both charge density and field, even at moderate values.

TABLE I .
Features of the models.

TABLE II .
Sets of parameters used for comparing models I and II ͑sets II.1 and II.2͒ and to fit to experimental data ͑set II.3͒.