1. Introduction
Charge carrier transport in disordered organics has been extensively studied for almost 50 years starting from 1970 [
1]. Early investigations in this field were confined to photoconductive materials using the time-of-flight (TOF) technique with pulsed photo excitation. It was discovered that the charge carrier transport in polyvinylcarbazole (PVK) and molecularly doped polymers (MDPs), widely used in electrophotographic industry, was
dispersive so that the post-flight TOF current decay was an algebraic (slow) rather than an exponential (fast) observed in crystalline solids [
2]. As a result, a mobility (called the drift mobility) should have been defined through an operational procedure by an intersection of the pre- and post-flight branches of a TOF curve presented in lg
j – lg
t coordinates [
3]. Continuous-time random walk theory of Scher and Montroll [
3] (a theory of choice in late 70s and early 80s) coexisted with a quasi-band multiple trapping (MT) formalism [
4,
5] but later both were superseded by the Gaussian disorder model (GDM) of the hopping transport developed by Bässler [
6]. All these results have been outlined in a book [
7]. The dipolar glass model (DGM) suggested in 1998 [
8] is surely to be noted.
Currently, the main interest of researchers shifted to a group of disordered organics employed in such electronic devices as light emitting diodes, photovoltaic cells, field-effect transistors and memory devices. Transport of charge carriers, mostly injected from electrodes, was now studied by using the finished parts of these devices trying to reproduce their operational characteristics by selecting model parameters. Besides, organic films were very thin (up to several nanometers) and the notion of the drift mobility as an averaged quantity became unproductive.
Common polymers are mainly used as dielectrics in electrical and electronic devices. For engineers, charge carrier mobility in polymers was not a main point of interest compared with their breakdown strength and durability under high electric stress. The situation changed with the advent of nuclear weapons/power stations and spacecraft technology. Needs of all these industries required developing a transport theory in common polymers because a simplistic accumulation of empirical data on the radiation-induced conductivity (RIC) systemized on the empirical level could not naturally satisfy community demand. Model development was greatly alleviated by the fact that time of flight (TOF) effects were usually minor compared with the recombination role.
In its contemporary form, the RIC theory employs the quasi-band multiple trapping formalism using an exponential trap distribution and became known as the Rose-Fowler-Vaisberg (RFV) model [
9]. Numerical calculations can also accommodate the Gaussian trap distribution with whatever values of energetic parameters for both trap distributions [
10].
An important turning point in these studies occurred during 2006–2017 when Tyutnev et al. (see our latest paper [
11]) developed a novel TOF technique; the so-called radiation induced variant employing pulses of 3 to 50 keV electrons provided by an electron gun ELA-65 (Orion, Moscow, Russia). In fact, this technique is a combination of three main modifications: TOF variant using 3 to 7 keV electrons producing carrier generation near an irradiated sample surface like in photo excitation; TOF-2 variant employing 50 keV electrons (the maximum energy available at the moment) for carrier generation in the bulk and TOF-1a method with the varying generation layer thickness. The main result of these investigations was that the carrier transport in MDPs was indeed moderately dispersive despite the fact that TOF curves in MDPs featured a horizontal plateau which in turn has been shown to be an artifact of a TOF technique (both photo- and radiation-induced) due to surface layers with a depleted dopant concentration.
For interpretation of these results, we used a conventional MT formalism with an exponential or the Gaussian trap distributions supplemented with appropriate computer codes. Eligibility of these MT models in describing hopping carrier transport in terms of the quasi-band theory seems to be solved positively based on a transport level concept which is currently under intensive development [
12].
But our recent investigations of the radiation-induced conductivity in commercial polymers (a legitimate representative of disordered organics) using pulsed and continuous step-function irradiations have shown that the above picture of the charge carrier transport described by the one-parameter exponential or the Gaussian distributions of hopping centers (respectively, traps) needs reexamination [
13]. Indeed, PVK and MDPs generally follow predictions of the conventional MT theories concerning both RIC and TOF experiments. The situation changes for films of polyethyleneterephthalate (PET), polyimide (PI), polystyrene (PS) and others. To describe their RIC, one needs using an aggregate trap distribution combining two exponentials as first suggested in the cited work.
In the present paper, we plan to carry out numerical calculations of both the RFV and the modified RFV (RFVm) models in a broad range of irradiation times and electric fields and to apply these results for interpretation of the published experimental data addressing some ambiguities.
2. Models Formulation
The RFV and RFVm models both employ the MT formalism whose basic equations are as follows:
Here, N is the total concentration of the mobile carriers (in our case, electrons). N0 is their concentration in conduction zone where they have the microscopic mobility μ0 and lifetime τ0. Energy trap distribution is M(E), total trap concentration is M0 (note that trap energy is taken to be positive). Distribution function of trapped carriers is given by ρ(E). Also, ν0 is the frequency factor, T-temperature, k-the Boltzmann constant. Here, variables N0, N and ρ are functions of time only and refer naturally to the RIC experiment. In a TOF setup, these variables become x-coordinate dependent (1D geometry is assumed).
Now, we have to specify the carrier generation/loss terms and formulate appropriate continuity equations. Below, the generation rate of quasi-free electrons is
g0 which is assumed to be constant and uniform during irradiation time
tg and is zero afterwards. A continuity equation in RIC may be written like this:
In a TOF setup when carrier exit to electrodes overrides recombination losses, it should be changed:
implying that now the concentration becomes necessarily coordinate-x dependent (
F0 is an applied field which is still uniform and constant). The generation rate has the previous meaning. As a consequence, TOF method becomes TOF-2 method.
Initial conditions for both setups are the same: at
t = 0 all concentrations are equal to zero. Boundary conditions for the TOF-2 case are specific and may be found in literature [
4,
14]. The output quantities of these calculations are the radiation-induced conductivity which is equal to
and the current density in a TOF-2 case (the concept of conductivity is not applicable here)
where
e-an elementary electric charge and
L-a film thickness.
At last, we have to specify a trap energy distribution. The conventional RFV model uses a simple exponential
where
E1 is in fact an average trap energy. Dispersion parameter α = kT/
E1 controls RIC current shapes for step-function irradiation in a small-signal regime [
4,
9,
14]. As mentioned earlier, for α ≤ 0.5 there are even closed-form expressions for these current shapes [
14]. In the RFVm model, the above trap distribution extends only to a separation energy
Es. For
E ≥
Es the distribution parameter of the second exponential
E2 rises appreciably. Now, we have to deal with two dispersion parameters: α
1 (the former α) and α
2 = kT/
E2. The idea of this separation is that each of two trap fractions clearly identifies its contribution to the RIC as suggested in References [
10,
13]. An explicit form of
M(
E) is as follows
where
For reference, we indicate that the relative fraction of deep traps is equal to
and for
we have
Here, M2 is the concentration of deep traps with energies exceeding Es. It should be noted that in the framework of the RFVm model as in the RFV one, the RIC prompt component (e is an electronic charge) is field dependent duplicating field dependence of g0 contrary to the genuine RIC prompt conductivity which is field independent. So, model parameters should be found by fitting experimental and numerical current curves of the RIC delayed components (below, RICd curves). In thin samples stressed by high fields, this approach converts into a TOF-2 setup.
5. Discussion
To our knowledge, there are no present-day studies of the RIC in polymers that aim to investigate its main characteristics in detail, if only vaguely reminiscent of our approach. Most publications report RIC data retrieved from experimental results relating to phenomena such as the electron charging of polymer films in which RIC could not be directly measured. Model of choice is a two-trap quasi-band formalism [
19,
20,
21] now extensively used by the ONERA French group of researchers [
22,
23]. Such data are intended to solve the urgent engineering problems but fail to contribute to an understanding of the detailed nature of the RIC phenomenon in polymers.
As noted in the Introduction, to deal with carrier transport in commercial polymers, we applied the radiation-induced TOF-2 variant which was initially developed to study charge carrier transport in photoconductive PVK and MDPs [
24,
25]. In this technique, we used a square (step-function) pulse which allows a time-resolved analysis of the build-up part of current transient in addition to the traditional processing of current decay after a pulse end. Exactly this approach combined with the small-signal RIC measurements on a broad time scale revealed limitations of the conventional MT theories to describe carrier transport in some commercial polymers [
13].
There is no direct relationship between the RIC and electron energy. In our practice, the standard electron energy is 50 keV but in the past we used 2 and 10 MeV electron beams. The RIC is generated by the time derivative of the absorbed energy converted into generation rate of charge carriers (electrons and holes). The real problem is to find an average absorbed energy over the whole film thickness (which depends critically on an electron energy and a sample thickness) and to relate it to the RIC using an appropriate model. Second, it is important to work with single pulses of electrons as pulse cycling leads to inevitable RIC degradation as a result of radiation damage to a test sample. In our practice, we had to study specifically pulse cycling effect and did it by distancing individual pulses by some minutes enough to make measurements of the RIC degradation.
To avoid unnecessary damage effects in RIC and transit time measurements, we worked only with fresh polymer samples for each irradiation run (exceptions are possible for pulse irradiations with total accumulated dose not exceeding 10 Gy [
9]. The prime aim of the RIC theory is to relate this time dependent energy profile with the local current density converging finally into the RIC current appearing in the external circuit.
6. Conclusions
The model of RIC in polymers developed in the present paper solves, for the first time, a problem of fundamental disagreement in the framework of the classical RFV theory between pulsed and continuous irradiations observed in some important commercial polymers such as PET, Kapton, polyethelenenaphthalene (PEN) while in some others including LDPE, PVK or MDPs this approach worked well (see
Figure 2 and
Figure 3) To achieve this, we introduced an aggregate two-exponential trap distribution accounting for deep traps with or without energy distribution allowing to describe unusual temporal behavior of the RIC under continuous irradiation. The physical origin of deep traps needs further clarification We developed a universal computer code and described how to fit experimental data by a tedious trial and error method. Our theoretical endeavor has become possible through almost 30 years of experimental RIC studies using an electron gun ELA-50 culminating in a remarkable paper [
13] in which the above mentioned fundamental contradiction has been made crystal clear. We conclude that the RIC in polymers must be studied under pulsed and long-time small-signal irradiation in laboratory conditions as described in Reference [
13] and should be interpreted in terms of the proposed RFVm (or RFV) models using appropriate computer codes to retrieve model parameters.
TOF effects predicted by the RFVm formalism are really extraordinary: they are almost nonexistent at weak fields and do appear only at very strong fields (see
Figure 1) when their explanation in terms of a drift mobility [
13,
15] is totally misleading, in which case they should be interpreted as a complex field effect described by the model presented in this paper.