Beyond the quasistatic approximation : Impedance and capacitance of an exponential distribution of traps

Electronic carriers in disordered organic and inorganic semiconductor materials, used in electronic, optoelectronic, and photovoltaic devices, are usually affected by an exponential distribution of localized states in the band gap traps . In this paper we provide a full solution of the relaxation of carriers in traps of such distribution, as a function of frequency and steady-state Fermi level. This includes in a unified treatment both the quasistatic limit, in which the traps modify time constants such as the trap-limited mobility, and the power-law relaxation at high frequency due to detrapping kinetics. We also analyze the combination of trapping with diffusion transport and recombination dynamics in photovoltaic devices. The different features of impedance and capacitance spectra are interpreted and also the analysis of the spectra in order to derive the main material parameters.


I. INTRODUCTION
Disordered organic and inorganic materials have raised increasing attention in the last two decades, in relation with applications in devices such as dye-sensitized solar cells ͑DSCs͒, 1,2 plastic solar cells, 3,4 organic light-emitting diodes ͑OLEDs͒, 5 and organic electronics. 6Frequency domain techniques are becoming of widespread used in the study of these materials and devices, since they provide an insight both on fundamental electronic processes and global device properties.Impedance and capacitance spectroscopy of devices 7,8 using such materials normally requires to consider a distribution of states in energy.Energy disorder is common for band-gap states of electronic carriers in disordered semiconductors, both for nanostructured metal oxides used in DSC ͑Refs.9 and 10͒ and for organic conductors used in thin-film transistors ͑TFT͒, [11][12][13] OLEDs, 8,14 and organic solar cells. 15specially in organic materials, deep traps are reactive sites as they are either radical cations or anions and can play a role in the degradation of organic molecules, such as polyphenylvinylene polymers.However, the degradation occurs at time scales of thousands of hours in the most stable systems.An important factor is therefore the presence of impurities which can also act as trapping sites, extrinsic traps, which can greatly influence the stability of the system.Also in ionic materials such as Li battery cathodes, energetic inhomogeneity has been found. 16he exponential distribution of states seems to play a key role in many of the mentioned systems.For example, such distribution is routinely obtained as a chemical capacitance in electrochemical measurements of nanostructured TiO 2 ͑Refs.17 and 18͒ and the exponential density of states ͑DOS͒ also explains the observed variation of electron diffusion coefficient in DSC using multiple trapping arguments. 9,10][14][15] Closely connected to such a distribution is the anomalous diffusion model 19 that so far seems to be the best approach for interpretation of diffusion of guest ions in transitionmetal oxides. 20,21st of the work presented in the literature for the interpretations of these systems uses a multiple trapping model, where the effect of traps is to reduce the mobility and increase the recombination time by a factor of the free to trapped charge.This approach, which dates back to Rose, 22,23 is formulated as a quasistatic approximation in Refs.24 and 25 and has been widely used in the DSC area. 10,26Similar approaches to trap-limited mobility are well known in the field of crystalline organic electronic devices 27 and amorphous organic semiconductors. 8,11This approximation is quite useful since it allows us to treat the trap distribution simply as an additional charge, which follows the relaxation of the free charge in the system, so that the time constants in the system can be easily deduced using the kinetic model of the trap-free system.However, any relaxation of the trapped carriers is assumed to be very fast and is neglected from the beginning.However, if the trap distribution indeed exists, one should be able to observe the relaxation of trappingdetrapping kinetics, provided that one measures in the appropriate frequency domain.A direct measurement of trap kinetics is very important for tailoring the materials with required properties for device operation.
There exist classical results on dispersive ͑power-law͒ behavior of the photoconductivity and mobility in time transient experiments, which is due completely to the trapping kinetics. 28,29Therefore, one can switch from the regime, where the traps remain in equilibrium, to that in which they dominate the transient response. 30In this paper we provide a full solution of the relaxation of the exponential distribution of traps, which unifies both regimes of behavior in a single framework, in the frequency domain.
It is well known that in semiconductor devices, each trap level provides a series RC circuit. 31The capacitance C trap is the chemical capacitance of a localized state 32 and relates to the equilibrium occupancy of the trap f ¯͑with 0 Յ f Յ 1͒ as It should be emphasized that C trap is related to the variation of the occupation of the trap with respect to the quasi-Fermi level.So this capacitance has nothing to do with a standard dielectric capacitance and is more related to the chemical capacitance, 17,33 also known as accumulation capacitance 34 or redox capacitance. 35On another hand the resistance R trap contains the kinetic factor for detrapping.Both values depend on the occupancy of the level, and the characteristic trap frequency is given by The value of t determines the relaxation properties of each level in the distribution.The properties of simple trap combinations in the frequency domain are illustrated in Appendix.We have discussed in detail in a previous paper the frequency-dependent capacitance in a two-state model, 36 and this model has been successfully applied in experiments of ion insertion in metal oxides. 16Using a similar method we provide here a full solution of the capacitance and impedance of the exponential DOS.Since the characteristic time for detrapping depends exponentially on the distance between a trap level and the transport level, the exponential distribution introduces a very large set of time scales.Furthermore, the trap response is largely variable depending on the occupancy of the trap.Therefore, it is required to establish the steady state of the DOS, at given Fermi level, as indicated in Fig. 1, and then to calculate the frequency-dependent capacitance C ‫ء‬ ͑͒, by integration of the capacitance of the different levels, and the associated impedance, Z͑͒ =1/ iC ‫ء‬ ͑͒.By summation, we give here a closed expression for the capacitance C ‫ء‬ ͑͒ of the whole distribution in a good approximation.
Very often, in experimental spectra, several frequency effects due to distinct phenomena ͑e.g., energy disorder and geometry inhomogeneity 16 ͒ have to be deconvoluted.For the interpretation of experimental data, it is very useful to have an understanding, in terms of analytical models, of the different spectral features in separate segments of the frequency window of measurement.Therefore, we will develop analytic approximations to the whole response of the exponential distribution of traps.The shape of spectra and different behaviors obtained at low and high frequencies will be discussed in detail, as well as the dependence of parameters on steady-state conditions, in order to assist the interpretation of experimental data.
The paper is divided into two main parts.Section II treats the relaxation of a continuous distribution of traps in the frequency domain, as shown schematically in Fig. 1.The goal of this section is to find out the local effects caused by the trapping-detrapping phenomena when all the states of the distribution are considered simultaneously.Section III describes an approach to spatially extended device modeling, by considering diffusion trapping in a restricted layer, and the effect of the trap distribution on recombination dynamics in the frequency domain, mainly in relation with photovoltaic systems such as DSC.We expect that these results will find application in a wider variety of systems.In this work we restrict our attention to systems in which the Fermi level can be considered homogeneous across the active device layer.We plan to apply the present developments in highly inhomogeneous systems such as TFTs and OLEDs in a future work.Another interesting development left for the future is to apply a similar method to the Gaussian distribution that is also widely used in organic semiconductors. 25,37

II. TRAP MODEL
Figure 1 shows a basic scheme of the model discussed in this paper.The scheme shows a piece of semiconductor, both with a transport level ͑extended states͒, and localized states in the band gap.The traps are assumed to be connected ͑by trapping-detrapping kinetics͒ to the transport level, which communicates with the outer contacts.In the scheme, we indicate that the applied potential governs the Fermi level ͑a specific realization of this is shown in Fig. 8; see also Ref. 2͒.Therefore, we assume that the local Fermi level can be fixed at a value E F .
The measurement involves two levels of perturbation.The first is the steady-state value of the potential.This fixes the occupation of the traps and the free-carrier concentration.More generally, if the traps also serve as recombination centers, they are normally not in thermal equilibrium with the conduction-band carriers, and they require separate Fermi levels. 38,39However, in the present work we do not treat such situation, and the traps are conservative with respect to number of carriers.The second level is a small harmonic perturbation.The latter causes the dynamic response of the traps and facilitates the analysis of experimental systems by looking at the spectral features of the capacitance and impedance.In essence the frequency response of a trap is a RC series circuit, as shown in Fig. 1 for two traps and discussed in Appendix.However, there is a great difference of trap response according to their steady-state occupancy.The empty FIG. 1. ͑Color online͒ The scheme shows a semiconductor where the Fermi level can be controlled with external potential.The semiconductor has both extended states ͑e.g., conduction-band lev-els͒ and localized ͑band-gap͒ levels.The Fermi-level marks approximately the separation between occupied and empty localized levels, which applies in an exponential distribution of traps.The equivalent circuit for two specific levels ͑one occupied, another one empty͒ is shown.For each level, the resistance corresponds to the thermal release of electron to the extended levels, and the capacitance corresponds to the variation of equilibrium occupancy of the level under variation of Fermi level, i.e., a chemical capacitance.The extended states also possess a chemical capacitance, though this is not shown.The characteristic frequencies of the equivalent circuits of occupied and empty levels have different properties, as discussed in the text, and are denoted as "deep" and "shallow," respectively.traps readily capture electrons, while the full traps do not.More exactly, the occupation of electron traps above the Fermi is well approximated by the Boltzmann distribution.Below the Fermi level, traps are almost fully occupied, but the density of vacant traps ͑holes͒ also obeys the Boltzmann distribution.The occupations are quantitatively shown in Fig. 2͑a͒ in the energy axis.The aim of the following development is to obtain the characteristic spectra of the combined relaxation of all the traps in the exponential distribution.

A. Kinetic equations
The model is formulated assuming the extended states at the level E c ͑conduction-band edge͒, with effective density N c and number density of carriers n c .With respect to the Fermi level E F , we have where k B is Boltzmann's constant and T the temperature.In addition, we assume a distribution of traps g͑E͒, with occupancy f t ͑E͒, where E is the energy; see the diagram in Fig. 2͑a͒ that illustrates the exponential distribution.The number of density of carriers in trap states is For the sake of brevity we omit the argument of f t in the following.The kinetic equations are Here, ␤ is the time constant for electron capture, usually given in terms of the thermal velocity of the electrons and capture cross section, In steady state, at given Fermi level E F , we have from Eq. ͑5͒ in terms of the Fermi-Dirac function, ͑8͒ By detailed balance condition, the time constant for release from the traps at the energy E t is

Frequency-dependent capacitance of a trap distribution
The method to obtain the frequency-dependent capacitance is explained in detail in Ref. 36 where q is the positive elementary charge.Note that all capacitances C in this section are given per unit volume.The extended states follow instantaneously the variation of applied voltage, which causes a variation of the Fermi level of the free electrons in the conduction band, −qdV = dE F .͑This assumption involves infinitely fast transport in the extended states and will be relaxed in Sec.III.͒Therefore, we can use the relationship This last equation can be expressed in terms of the chemical capacitance of the extended states, 17,33 C 0 = q dn ¯c dV ¯= q 2 k B T n ¯c.͑13͒ In contrast to Eq. ͑11͒, the traps do not follow instantaneously the voltage variation, since each trap only communicates with the conduction band.By solving Eq. ͑5͒ for a small perturbation, we obtain where the trap frequency has the expression From Eqs. ͑12͒ and ͑14͒ we obtain that the trap occupancy depends on the frequency as Inserting Eqs.͑13͒ and ͑16͒ in Eq. ͑10͒, we obtain The second term in Eq. ͑17͒ is the frequency-dependent capacitance of the trap distribution.It has the form where C t ͑E͒ is the equilibrium chemical capacitance per unit energy, 32 so that C t dE is the capacitance associated to the traps in the interval dE.Equation ͑18͒ means a parallel connection of all the separate capacitances of the traps, as indicated in Fig. 3. Integrating Eq. ͑14͒ over the energy axis, we can write the relationship between ac modulation of trapped and free carriers as From Eqs. ͑18͒ and ͑19͒ a numerical calculation of the frequency-dependent trap capacitance, at given Fermi level, is now possible and will be used below to plot the exact results.As indicated above, we also wish to understand the separate features of the spectra in different ranges of the frequency domain.According to Eq. ͑9͒ the levels below E F are almost fully occupied, while the levels above E F are populated following the Boltzmann distribution: . This makes a great difference for the kinetics of the traps; therefore, a main distinction in the calculation below is made for trap levels above and below the Fermi level.We have denoted them, respectively, as "shallow" and "deep" traps, as indicated in Fig. 1.This is a convenient notation for the purpose of the model that does not imply that either kind of trap is close or far from the transport level in an absolute sense.In addition, the Boltzmann statistics can also be used to describe vacant states below the Fermi level.The use of the Boltzmann statistics, to describe the trap occupancy in suitable domains of the energy axis, is a good approximation for the exponential distribution and other distributions.However, in other cases, such as the Gaussian distribution when the Fermi level is deep, this approach fails.Indeed the carrier occupation at low carrier density has the form of a Gaussian distribution around d 2 / k B T below the center of the DOS ͑Ref.37͒ ͑see Ref. 25 for a discussion of this question͒.Thus, the Gaussian distribution requires a separate treatment.

C. Characteristic trap frequencies
Let us discuss the dependence of characteristic frequencies ͓Eq.͑15͔͒ on the trap energies using approximations to the Fermi-Dirac distribution f ¯t.For E t Ͻ E F , the trap frequency is constant, On another hand, for E t Ͼ E F the trap frequency increases with E t , These results are illustrated in Fig. 2͑b͒ and are independent of the distribution g͑E͒, provided that the approximation of the Boltzmann distribution can be used, as already mentioned.The different dependence of t below and above E F makes it a useful step to split the integral in Eq. ͑18͒ in two parts,

D. Capacitance of deep levels
According to Eq. ͑21͒, all the occupied sites bellow E F ͑which we will call the deep levels͒ relax with the same frequency, t0 , which is also the slowest frequency in the distribution.We can immediately infer that the deep levels can be described as a RC series circuit, The constant capacitance in Eq. ͑24͒ consists simply on the summation of the chemical capacitances over all the levels, The resistance in the equivalent circuit of Eq. ͑24͒ is ͑see Appendix͒ Since t0 is the slowest frequency in the distribution, we deduce that the observation of the trap relaxation in the impedance measurement will occur only if the measurement frequencies are Ͼ t0 .If, on the other hand, the measurement is such that Ͻ t0 , Eq. ͑18͒ reduces to a constant capacitance, the chemical capacitance of the localized levels, In this case, the traps and extended state occupation remain in equilibrium throughout the measurement, so there are no kinetic manifestations of the traps.This forms the basis for the quasistatic approximation to multiple trapping systems, which will be shown in detail in Sec.III, below.
With respect to the shallow levels, at E t Ͼ E F , we note in Eq. ͑22͒ and Fig. 2͑b͒ that t increases exponentially with energy, up to the maximum value Therefore, the relaxation of these levels, given by has to be calculated for the specific distribution of traps.In the following we calculate the two components of the frequency-dependent capacitance, for deep and shallow levels, in the specific case of an exponential distribution defined as Here, N t is the total volume density of localized states in the distribution, k B is Boltzmann's constant, and T 0 is a parameter with temperature unit that defines the depth of the distribution below E c .

E. Relaxation of an exponential distribution of traps: Deep levels
For the deep levels we note that the following approximation is valid: Calculating the integral in Eq. ͑25͒ we obtain and from Eqs. ͑21͒ and ͑26͒ we get the resistance dependence on Fermi level

Relaxation of an exponential distribution of traps: Shallow levels
For the shallow levels we find Let us introduce the exponents

͑36͒
In a previous work, 36 by using a distribution with certain properties concerning the characteristic frequencies, it was shown that one can obtain the constant-phase element ͑CPE͒ behavior, with impedance Z͑͒ = Q −1 ͑i͒ −m .We treat the integral in Eq. ͑36͒ as described in Ref. 36.Using the change of variables t = i tm −1 e −͑E−E c ͒/k B T , t 0 = i / t0 , and t m = i / tm , we obtain In the range of frequencies t0 Ӷ Ӷ tm , we can set in the integral in Eq. ͑37͒ t m Ϸ 0 and t 0 Ϸ ϱ.The final results are Equation ͑38͒ represents a CPE element and dominates the capacitance C traps ‫ء‬ at t0 Ӷ Ӷ tm .The reduction of the combination of the traps to a simple CPE model is indicated in Fig. 3.The CPE behavior of the impedance is, according to Eq. ͑38͒, By Laplace transform to the time domain, Eq. ͑40͒ is directly related to the classical results on dispersive ͑power-law͒ behavior of the mobility in transient experiments 29 in disordered semiconductors.
Figure 4 illustrates the behavior of the trap capacitance as a function of the frequency.In the representation, we denote CЈ =Re͑C ‫ء‬ ͒ and CЉ =Im C ‫ء‬ .It is observed that the series circuit of R deep and C deep0 is a good approximation to the exact capacitance in the low-frequency range ͑ Ͻ t0 ͒.It is also observed that the CPE in Eq. ͑38͒ is an excellent approximation to the exact result in the high-frequency range ͑ t0 Ͻ Ͻ tm ͒.
Figure 5 shows the capacitance and impedance in the complex plane representation.͑Note that the elementary trap resistances are connected in parallel, hence the macroscopic resistance of a film is inversely proportional to the volume of material.͒At low frequencies ͑below t0 ͒, the impedance behaves indeed as a series circuit of R deep and C deep0 .In fact, it is observed in Fig. 5͑a͒ that R deep and C deep0 provide a good approximation to the dc value ͑ =0͒ of the impedance R traps ͑0͒ =Re͓Z͑0͔͒ and capacitance C traps = C traps ‫ء‬ ͑0͒, respectively.These parameters are represented as a function of Fermi level in Fig. 6.Note that Eqs.͑32͒ and ͑33͒ introduce a factor of error to the exact dc values of the exponential distribution.The reason for this is clear in Fig. 5͑a͒: the values R deep and C deep0 provide the low-frequency arc in the capacitance plot, and therefore a contribution to the capacitance by the shallow levels, at higher frequency, is neglected.Nonetheless, Eqs.͑32͒ and ͑33͒ constitute useful approximations with analytically closed expressions and describe well the Fermi-level dependence of R traps ͑0͒ and C traps , as seen in Fig. 6.
Figure 7 shows in more detail the spectral features of the impedance.In Fig. 7, the time constant for electron capture has been decreased with respect to Fig. 5, and we see that this modification has a large impact on the value of the trap resistance.The overall shape is similar to the simple two-trap model in Fig. 14͑b͒, with a transition from the parallel resistance value observed at the intercept in Fig. 7͑a͒ to the lowfrequency value in Fig. 7͑c͒.The additional feature, introduced by the exponential distribution, is the CPE behavior, a tilted straight line of slope ␣ in the intermediate frequency range, which is observed in Fig. 7͑b͒.

III. DEVICE MODELING
Models for electronic ͑or ionic or mixed͒ devices are formulated with the appropriate transport and/or conservation equation for each kind of carrier.Models can vary widely depending on the number of material layers, different phases in each layer, and the boundary conditions that describe carrier transfer or blocking at the different interfaces.In Sec.II, we have formulated the trap model in the frequency domain using the standard approach that assumes the same average distribution of electronic levels at each point of the phase that shows such distribution.Therefore, whatever the device model, the presence of the exponential distribution implies that the development of Sec.II "hangs" at each point as part of the conservation equation, in addition to the specific transport features of the device model.
Obviously, device model can become quite complicated, especially in those cases in which the carrier distribution is heavily inhomogeneous.The trap occupation will vary from point to point, and this usually requires numerical solution.In order to illustrate the application of the trap model, we consider in this section the model of transport by diffusion coupled with trapping in a layer of thickness L. This model is not trivial and applies in a variety of situations, but it has the advantage that it is a one-dimensional problem in which steady-state carrier distribution is nearly homogeneous and can be solved analytically.
An example of application is presented in Fig. 8.The scheme shows a nanostructured and undoped metal-oxide semiconductor, deposited on a conducting substrate ͑left boundary͒, and immersed in electrolyte solution that compensates with ions any increase of electronic charge in the solid porous phase. 9Electrons can be injected and extracted in the film from the substrate, diffuse in the film, and are reflected at the outer edge ͑right boundary͒ of the semiconductor layer.This structure is of interest for a number of nanostructured electrochemical devices, 9 among which the most prominent are the photoelectrochemical solar cells such as DSC. 1,2It is well documented that metal oxides such as TiO 2 used in DSC exhibit an exponential distribution of electronic traps, 18 and such distribution exerts an important influence over the electron diffusion coefficient.This subject has been reviewed recently 25 using the quasistatic approximation, and in the following we will develop the complete dynamics of traps in the frequency domain.
The observation of trap dynamics in electron transport in nanostructured TiO 2 has been reported in the time domain at very short-time scales, 40,41 but to our knowledge, the ample body of experimental data in the frequency domain ͑see Ref. 7 and references therein͒ has not been scanned in search of trap dynamics, very probably due to the lack of suitable models that expose the frequency response of the trap distribution.It is also likely that the dynamics of electronic traps is far too fast to be observed in the usual frequency range of impedance spectroscopy ͑0.1 Hz-1 kHz͒.In the case of ionic solid conductors, such as electrochromic and Li battery materials, the trap dynamics is generally much slower and more likely to be clearly detected.So one important point in the following model is to justify the appearance of anomalous diffusion in the case of the exponential distribution of traps.Such model was developed using formal considerations, 19 and it has been found very powerful for describing electrointercalation diffusion in solids. 21,42Recently, published impedance data 43 seem to realize all the features of the diffusion-trap model, including anomalous diffusion features.
Another central aspect of photovoltaic applications, coupled with diffusion, is recombination, and this will also be considered below.In general, recombination means the loss of electronic carriers, by meeting another kind of carrier, as in electron-hole annihilation.In contrast to trapping process, in which the carriers are accumulated in certain localized states and can be retrieved, recombination is an irreversible process where the carriers are lost by transformation, and therefore, the free energy associated with such carriers does not contribute to energy output of the solar cell.
The general situation treated in this section is illustrated in Fig. 8. Injection ͑or extraction͒ of carriers causes diffusion along the extended states.While they advance, free carriers have the chance to be captured, and later released, by traps.In addition, free carriers have a possibility to be captured in recombination process.The recombination process specifically shown in Fig. 8 is the electron transfer from the metaloxide nanoparticles toward ionic species in solution ͑these are depicted around their own Fermi level, the redox poten-tial͒.This is the dominant recombination mechanism in standard liquid electrolyte DSC, 1 where the I − / I 3 − redox couple is normally used to regenerate the oxidized dye molecules from the counterelectrode.The concentration of electron acceptor ions is very high and is not affected by charge transfer from the semiconductor.The exponential distribution of traps has been amply applied in the studies of DSC with a liquid redox electrolyte, and it is agreed that the lifetime dependence on Fermi level is well explained by the trapping effect; 10,44,45 however, these studies all used the quasistatic approximation.We should also remark that DSC can be made also with organic electronic hole conductors filling the voids of the porous network, and the interfacial charge transfer implies electron-hole recombination. 46n Secs.III A-III C we derive first the theory of impedance for diffusion trapping in spatially restricted conditions and, thereafter, the model for recombination coupled with trapping.The main point is to reveal the effects of trap dynamics on diffusion and recombination phenomena beyond the quasistatic approach that was used in previous works. 24,25,45,47

A. General features of the diffusion impedance
Considering the different processes outlined in Fig. 8, we extend Eq. ͑4͒ to the transport-conservation equation as follows: In Eq. ͑41͒ 0 is the lifetime of electrons in the extended states and J n is the flux of free carriers at position x that relates to the gradient of concentration by Fick's law, where D 0 is the diffusion coefficient of the free electrons in extended states.The analysis of diffusion impedance models has been presented in previous papers 19,48,49 and we build on those results here.The solution of Eq. ͑41͒ for small ac perturbation, with a blocking boundary condition, is the general expression of the diffusion impedance in a film of thickness L, Here s = i and r m is the transport impedance per unit length per area ͑⍀ m͒, where R 1 M is the macroscopic transport resistance of the film of area A. r m is the reciprocal to the electronic conductivity n ,

͑45͒
The conductivity relates to the free electron diffusion coefficient as where C 0 is the specific chemical capacitance of the extended states 48,50 defined in Eq. ͑13͒.Equation ͑46͒ is an expression of the Einstein relation. 25We note that the conductivity is given by The element m in Eq. ͑43͒ adopts different forms, depending on the local processes included in the model ͑trapping, recombination, etc.͒.For the interpretation of the model, it is useful to represent Eq. ͑43͒ as a distributed equivalent circuit ͑a transmission line͒.The general impedance in Eq. ͑43͒ is represented in Fig. 9͑a͒, and the transmission lines ͑b͒-͑f͒ represent specific models that will be discussed in this section.Note that Fig. 9͑a͒ must be regarded as a continuous model in which the branching does not correspond to finite distances.The elements r m , associated with diffusive transport, 50 are distributed in the spatial direction and are continuously interrupted by transversal elements m .This combination corresponds to the physical probabilities for electronic events in Fig. 8 and gives rise to the characteristic structure of a transmission line.Generally, linear equations for physical quantities varying in space with local dissipation are represented with transmission lines, for example, acoustic waves. 51e discuss first the simplest ͑ordinary͒ diffusion model, with no traps and no recombination, shown in Fig. 9͑b͒.In this model the distributed admittance m −1 consists on the chemical capacitance Here, C 0 M is the macroscopic chemical capacitance of the film.Equation ͑43͒gives The characteristic transport frequency d0 is the reciprocal of the transit time through the layer of thickness L, The impedance spectrum of blocked diffusion is shown in Fig. 10͑a͒.At high frequencies the spectrum exhibits the 45°l ine ͑Warburg impedance͒ associated with diffusion.At frequencies lower than d0 the spectrum is capacitive, and the low-frequency resistance is R 1 M / 3.This is an important feature, since the electronic conductivity of the semiconductor layer can be directly extracted from R 1 M , by Eqs.͑44͒ and ͑45͒.
For diffusion and recombination, there appears a recombination resistance in parallel with the chemical capacitance in the transmission line ͓see Fig. 9͑c͒ and Ref. 48͔.This will be further discussed in Sec.III C. The transverse impedance in this case is The characteristic frequency of recombination is rec0 = 0 −1 , ͑52͒ and the distributed recombination resistance is given by where R 3 M is the macroscopic recombination resistance of the layer.In contrast with Fig. 10͑a͒, the spectra of diffusion recombination, shown in Figs.10͑b͒-10͑d͒, are resistive at low frequencies.There are two competing processes now, the transport across the layer ͑ d0 ͒ and the carrier loss by recombination ͑ rec0 ͒.The shape of the spectra is regulated by the factor relating the characteristic frequencies, which can be expressed in several alternative ways, 48 where L n = ͱ D 0 0 is the electron diffusion length.In the case of low recombination ͑ rec0 Ӷ d0 or L n ӷ L͒, the carriers have plenty of time to cross the layer before they are lost.The spectrum is a recombination arc, shown in Fig. 10͑b͒, and further discussed below.The diffusion Warburg is a small feature in the high-frequency part, clearly visible in the enlarged plot of Fig. 10͑c͒, where we can see the start of the curvature of the capacitive part of Fig. 10͑a͒, due to recombination.In the spectrum of Fig. 10͑c͒ we can still determine the electronic conductivity from R 1 M / 3, i.e., the resistance at the turnover, and this method has been effectively applied in experiments. 7,52n the other hand, in the case of strong recombination ͑ rec0 ӷ d0 or L n Ӷ L͒ the injected carriers penetrate a restricted extent in the layer, and the boundary condition is irrelevant.The recombination resistance distorts the diffusion spectrum even at frequencies higher than the frequency of turnover d0 , and the spectrum adopts the form of the Gerischer impedance, shown more clearly in Fig. 10͑d͒, which corresponds to diffusion recombination in semiinfinite space.

B. Diffusion-trapping impedance
The calculation of the impedance of diffusion and trapping is given in Ref. 49 for a single trap.The result is that the transverse element in Eq. ͑43͒ takes the form The resulting transmission line is shown in Fig. 9͑d͒.According to Eq. ͑55͒, the RC circuit of the trap is connected in parallel to the capacitance C 0 of the extended states.
Of interest here is the diffusion-trap impedance for an exponential distribution of localized states.It can be easily shown that the ac impedance, obtained using Eqs.͑41͒ and ͑5͒, has the form of Eq. ͑55͒, where now, C traps ‫ء‬ is the full capacitance of the distribution ͓Eq.͑18͔͒, which we have characterized in Sec.II.The distributed equivalent circuit is shown in Fig. 9͑e͒, where the transverse element should be regarded as that given in Fig. 3.So we can readily calculate the shape of the spectra by including Eq. ͑55͒ in Eq. ͑43͒.
A first set of results is shown in Fig. 11.In this example, the lowest trap frequency t0 is larger than the frequency of transit time dn .Therefore, the overall shape of the impedance spectra, shown in Fig. 11͑a͒, is the same as that of diffusion without traps, shown in Fig. 10͑a͒.The kinetic effect of trapping appears as a modification of the Warburg domain, at high frequencies, shown in Figs.11͑b͒ and 11͑c͒.We find two interesting features in this high-frequency range.First, at the highest frequencies, Fig. 11͑c͒, we observe the Gerischer impedance, associated with diffusion recombination, and shown before in Fig. 10͑d͒.This, in fact, occurs at frequencies higher than tm , which is for the fastest trap, i.e., the upper cutoff frequency of the distribution.The appearance of the Gerischer impedance is due to the fact that above tm , the trap capacitors are shunted, hence the trap behaves like a pure resistance, and the circuit in Fig. 9͑d͒ reduces to that in Fig. 9͑c͒.
In the intermediate frequency range ͑ Ͻ tm ͒, we observe in Fig. 11͑c͒ a diffusion line with slope lower than 45°.This indicates the occurrence of anomalous diffusion, 19 and this is caused by the reduction of Z traps to a CPE, as pointed out in Fig. 3 and discussed in Sec.II F. The CPE replaces the pure capacitance C 0 of ordinary diffusion, and the transmission line of Fig. 9͑e͒ adopts, in a restricted frequency range, the form of Fig. 9͑f͒, which is the anomalous diffusion model. 19,53et us return to the global spectrum shown in Fig. 10͑a͒.Although it has the form of ordinary diffusion, as already commented, there is a strong change of the characteristic frequency of turnover, dn , with respect to that of free diffusion, d0 .This is clearly appreciated in the representation of the impedance as a function of the frequency ͓Fig.11͑d͔͒, Z' (kΩ) FIG.11. ͑Color online͒ Representation of the impedance of diffusion in a film of thickness L, with an exponential distribution of traps.͓͑a͒-͑c͔͒ Impedance spectra in the complex plane.The gray thin line is the impedance of diffusion without traps.The yellow square points mark the diffusion frequency dn = ͑C 0 / C traps ͒ d0 , the cutoff frequency tm , and the deep traps frequency t0 .͑d͒ The real part of the impedance as a function of the frequency.The parameters used in the calculation: T = 300 K, E F = 0.75 eV, E c = 1 eV, T 0 = 1000 K ͑␣ = 0.33͒, N c =10 18 cm −3 , N t =10 20 cm −3 , D 0 =10 −4 cm 2 s −1 , L =10 −3 cm, and A =1 cm 2 .
where both models are shown, for diffusion with and without traps.Here it is seen that the frequency of turnover is lowered by five decades of frequency.The reason for this is that, even though the trap relaxation occurs only at high frequencies, the traps strongly increase the equilibrium capacitance in the low-frequency range, which is given by C traps ͓Eq.͑27͔͒, as mentioned in Sec.II D. Therefore, the turnover frequency from the Warburg to the capacitive part has the value Comparing with Eq. ͑50͒, we observe that the measured diffusion coefficient in the presence of traps is In general, D n can be interpreted as a chemical diffusion coefficient. 25,47Note that the measured conductivity ͑from R 1 ͒ is not modified, since n is the same as that in Eq. ͑46͒, Equation ͑58͒ corresponds to the generalized Einstein relation. 25hese results are an application of the quasistatic approximation 24 that was mentioned in Sec.I.If the trapping dynamics is fast, then the system can be described in terms of the transport model of the trap-free system.There is a modification of the measured time constant, i.e., the chemical diffusion coefficient, which is reduced by the additional capacitance of the traps.However, the dc quantities, such as the dc conductivity, are not modified at all. 25 In Fig. 12 we plot the impedance of diffusion trapping with another set of parameters and we obtain a very different result.The change is due to the fact that trapping kinetics, in this case, are slower than the transit time.Thus, the shape of spectra ͓Fig.12͑a͔͒ is not that of diffusion, but instead, we obtain the impedance response of the exponential distribution of traps, shown in Figs. 5 and 7.As before, at high frequency we obtain the Gerischer impedance, which is the manifestation of diffusion trapping.In Fig. 12͑c͒ we observe that the low-frequency resistance is mainly due to traps in this case and not associated to the transport resistance.Therefore, the quasistatic approximation is invalid in this case, and we cannot infer the conductivity from the lowfrequency resistance, in contrast with the previous example.

C. Trapping-recombination impedance
From previous general analysis of the impedance of solar cells, we know that recombination introduces a resistance in parallel to the chemical capacitance. 54This has been already remarked above in Fig. 10͑b͒.In the case in which the diffusion transit time, L 2 / D 0 , is much faster than the electron lifetime 0 , the impedance response is basically given by a recombination arc.In the following, we wish to calculate the modifications of the recombination impedance by the effect of the exponential distribution of traps.The result we expect is a resistance in parallel with the full capacitance C ‫ء‬ ͑͒ of the DOS, related to both the extended states and exponential distribution.
In order to show this, we consider that diffusive transport is very fast in the system of Fig. 8.This implies that carrier densities are homogeneous.Integrating Eq. ͑41͒ between the extracting boundary at x = 0 and the blocking boundary at x = L, we obtain For a small perturbation in the frequency domain, Eq. ͑59͒ gives Using Eq. ͑20͒, we can write Eq. ͑60͒ as where the recombination resistance is defined in Eq. ͑51͒.
The impedance of the film, per unit area, is 12. ͑Color online͒ Representation of the impedance of diffusion in a film of thickness L, with an exponential distribution of traps.͑a͒ and ͑b͒ show the impedance in the complex plane.The gray thin line is the impedance of diffusion without traps.The yellow square point marks the cutoff frequency tm .͑c͒ shows the real part of the impedance as a function of the frequency.The parameters used in the calculation: T = 300 K, E F = 0.75 eV, E c = 1 eV, T 0 = 1000 K ͑␣ = 0.33͒, N c =10 18 cm −3 , N t =10 20 cm −3 , D 0 =10 −4 cm 2 s −1 , L =10 −3 cm, and A =1 cm 2 .

Z =
This shows, as anticipated, that recombination introduces the resistance r rec in parallel to the total frequency-dependent capacitance C ‫ء‬ ͑͒ of Eq. ͑17͒.
Let us assume that the frequencies of the measurement are lower than the frequency t0 .In this case Eq. ͑20͒ reduces to where C traps is given in Eq. ͑27͒.Equation ͑63͒ is the rigorous formulation of the quasistatic approximation 24 for small modulation quantities.It means that the modulation of the trapped charge follows instantaneously the modulation of the free charge, as mentioned before, and the ratio of these quantities is given by equilibrium chemical capacitances of the corresponding states.If Eq. ͑63͒ holds, then from Eq. ͑62͒ we arrive at the impedance and this can be written where, by Eq. ͑53͒, Equation ͑64͒ means that if Ӷ t0 the relaxation dynamics is the same as that of free electrons. 54There are no kinetic effects of traps.The traps, however, modify the time constant for recombination by the ratio of equilibrium capacitances ͓Eq.͑66͔͒.This is because displacing the Fermi level requires to discharge the traps, as was first noted by Rose. 22,23he quasistatic approximation considerably simplifies the analysis of experimental techniques that probe the properties of DSC; see, for example, Refs. 10 and 45 that discuss the experimental determination of Eq. ͑66͒.
In Fig. 13 we have plotted the impedance of the exponential distribution of traps with the parallel recombination resistance ͓Eq.͑62͔͒.We have neglected the free charge capacitance C 0 since this is usually not significant in comparison to C traps ‫ء‬ , unless E F is very close to the conduction-band edge.Now there are two characteristic frequencies in the system: t0 indicates the onset of trap relaxation and CPE behavior, as discussed above, and rec = n −1 is the effective lifetime frequency.
In Fig. 13͑a͒ we have t0 ӷ rec , therefore the traps relaxation discussed in Fig. 7, occurs undisturbed at very high frequencies.When lowering the frequency, the impedance closes toward the real axis by the effect of recombination resistance, as in Fig. 10͑b͒.The recombination arc 54 has the characteristic frequency n −1 .In Fig. 13͑b͒ we have still t0 Ͼ rec , but their values are closer than in the previous example, and in this case the trap relaxation is visible as a modification of the recombination arc in the high-frequency part.Finally, in Fig. 13͑c͒, we have t0 Ͻ rec .In this case the recombination resistance is smaller than the trap resistance.Therefore, the CPE behavior of trap relaxation is fully visible and is converted into an arc at the lower frequencies by the parallel recombination resistance.
It should be recalled that Fig. 13 represents a homogeneous model of recombination, in which carrier transport is infinitely fast, as assumed above.The impedance patterns of Fig. 13 are remarkably similar to those of the diffusionrecombination impedance in Figs.10͑b͒ and 10͑c͒.In principle, this similarity introduces an ambiguity of models that could lead to mistake interpretation of trapping recombination ͑or trapping reaction͒ as diffusion-recombination results.However, from the Fermi-level dependence of the resistance in the high-frequency straight part, both models, diffusion and traps, should be readily distinguished.Indeed, in the diffusion model the high-frequency feature gives the transport resistance R 1 , which depends on the potential as exp͓͑E c − E F ͒ / k B T͔ ͓Eq.͑47͔͒.In contrast, the Fermi-level dependence of the traps resistance is much more steep, as indicated in Eq. ͑33͒ and in Fig. 6͑b͒.Therefore, we should emphasize that the spectra should be monitored over a sufficient variation of bias potential in order to determine the correct interpretation.In experimental results of impedance of DSC, the mentioned transport resistance dependence of Fermi level was observed, 7,55 it therefore appears that trap relaxation is quite fast in nanostructured TiO 2 surrounded with liquid electrolyte, and the widespread application of the quasistatic approximation is well justified.
The extension of the approach presented in this section, for nonhomogeneous device models, requires a twofold numerical solution.First, steady-state equations must be solved to find the local values of charge density at each point of the device.Then, such information allows us to write the transport equations for small ac quantities, which contain also dc parameters.An example is given in Ref. 50.

IV. CONCLUSIONS
The relaxation of an exponential distribution of traps in the frequency domain contains two main features.At low frequency the distribution gives the normal relaxation of a single trap, i.e., a RC series circuit.At high frequency the distribution responds as a constant-phase element.The CPE exponent is independent of the Fermi-level position but changes with the temperature as ␣ = T / T 0 .The transition between the two features occurs roughly at the frequency of the highest occupied trap level at the position of the Fermi level.In the presence of diffusion and recombination, the interplay between the frequency of the highest occupied trap level and the effective ͑chemical͒ diffusion coefficient or effective lifetime determines the shape of impedance spectra.In the presence of very slow traps, the quasistatic approximation is not valid, since the low-frequency response is governed by the trap kinetics.We have found similarities between diffusionrecombination and trapping-recombination impedance models.Therefore, a careful analysis of experimental data is necessary, regarding these models, and exploring a wide window of potential and if possible temperature seems recommended to identify correctly the trap relaxation by impedance spectroscopy.The variation of impedance parameters with bias potential and temperature in agreement with the model should provide strong evidence for the identification of the trap kinetics.

APPENDIX: IMPEDANCE AND CAPACITANCE OF SIMPLE COMBINATIONS OF TRAPS
The frequency-dependent capacitance of a single trap is . ͑A1͒ Here, C t1 is the trap capacitance and t1 is a constant, the characteristic trap frequency.The impedance associated with Eq. ͑A1͒ is where the trap resistance is defined as From Eq. ͑A2͒ we see that the equivalent circuit of the trap is RC series.In the complex plane the capacitance forms an arc, of width C t1 , while impedance forms a vertical line, being the resistance constant at all frequencies.This is illustrated in Fig. 14.

͑A5͒
In the high-frequency limit this gives Therefore, the impedance intersects the x axis at the value of the parallel combination of the trap resistances.This is because the impedances of the capacitors vanish at high frequency.On the other hand, the low-frequency limit of the real part of the impedance depends on the capacitances ͑via the characteristic frequencies͒,

͑A7͒
The impedance rises vertically when the frequency decreases, but it makes a transition between the values in Eqs.͑A6͒ and ͑A7͒ ͓see Fig. 14͑b͔͒.*bisquert@fca.uji.es

FIG. 2 .
FIG. 2. ͑Color online͒ ͑a͒ Energy diagram showing the thermal occupation at temperature T = 300 K of an exponential DOS in the band gap.E F is the Fermi level of electrons and E c is the conduction-band energy.The clear ͑green͒ shaded region indicates the band-gap states that are occupied with electrons, and the dark ͑blue͒ shaded region indicates empty levels, as determined by the Fermi-Dirac distribution function ͑red dashed line͒.͑b͒ The trap frequencies as a function of energy.The parameters used in the calculation: T 0 = 800 K, N c =10 20 cm −3 , N t =10 20 cm −3 , and ␤ =10 −10 cm 2 s −1 .

FIG. 3 .
FIG. 3. ͑a͒The equivalent circuit of the conduction-band states ͑represented by capacitance C 0 ͒ and the distribution of traps, each energy interval dE being associated with a RC circuit.͑b͒ The trap impedance is represented by Z traps .͑c͒ Reduction of the impedance of an exponential distribution of traps Z traps to a much simpler equivalent circuit, valid in the high-frequency domain, as is demonstrated quantitatively in the text.The sum of individual RC elements, each corresponding to a narrow energy interval, over all the shallow levels ͑above the Fermi level͒ collapses into a CPE element.

FIG. 4 .FIG. 5 .
FIG. 4. ͑Color online͒ Representation of the real and imaginary parts of the frequency-dependent capacitance C traps ‫ء‬ for an exponential distribution of traps as a function of the frequency.Also shown are the approximations at low frequency ͑C deep ‫ء‬ ͒, an RC circuit, and high frequency ͑C shallow ‫ء‬ ͒, a CPE element.The square point marks the deep trap frequency t0 .The parameters used in the calculation: T = 300 K, E F = 0.4 eV, E c = 1 eV, T 0 = 800 K ͑␣ = 0.375͒, N c =10 20 cm −3 , N t =10 20 cm −3 , and ␤ =10 −10 m 2 s −1 .

FIG. 6 .FIG. 7 .FIG. 8 .
FIG.6.͑Color online͒ The thick red lines show the dc value of the ͑a͒ capacitance and ͑b͒ resistance for an exponential distribution of traps as a function of the Fermi level.In blue dashed lines are shown the dc values for the capacitance and resistance of the deep levels, according to the approximations discussed in the main text.The parameters used in the calculation: T = 300 K, E c = 1 eV, T 0 = 800 K ͑␣ = 0.375͒, N c =10 20 cm −3 , N t =10 20 cm −3 , and ␤ =10 −10 cm 2 s −1 .

FIG. 9 .FIG. 10 .
FIG. 9. ͑a͒ Transmission line representation of the diffusion impedance with the distributed diffusion resistance r n and the general transverse element n .͓͑b͒-͑f͔͒ Specific models are indicated, depending on the local electronic processes coupled with diffusion.
FIG.12.͑Color online͒ Representation of the impedance of diffusion in a film of thickness L, with an exponential distribution of traps.͑a͒ and ͑b͒ show the impedance in the complex plane.The gray thin line is the impedance of diffusion without traps.The yellow square point marks the cutoff frequency tm .͑c͒ shows the real part of the impedance as a function of the frequency.The parameters used in the calculation: T = 300 K, E F = 0.75 eV, E c = 1 eV, T 0 = 1000 K ͑␣ = 0.33͒, N c =10 18 cm −3 , N t =10 20 cm −3 , D 0 =10 −4 cm 2 s −1 , L =10 −3 cm, and A =1 cm 2 .

FIG. 13 .
FIG.13.͑Color online͒ Repesentation of impedance of a film of thickness L, with an exponential distribution of traps with recombination from the conduction band.The green square point marks the deep trap frequency t0 , and the blue round point marks the effective lifetime frequency n −1 , with the values ͑a͒ 0 = 1 s and n = 362 s, ͑b͒ 0 = 0.01 s and n = 3.62 s, and ͑c͒ 0 = 0.001 s and n = 0.362 s.In ͑c͒ in green dashed line the CPE approximation at high frequency is indicated.The parameters used in the calculation: T = 300 K, E F = 0.4 eV, E c = 1 eV, T 0 = 800 K ͑␣ = 0.375͒, N c =10 20 cm −3 , N t =10 20 cm −3 , and ␤ =10 −10 cm 2 s −1 .