Using static current models for the free current

This exclusive separation has also been used by Jonscher [78, pp. 40-41]. He associated the dc conductivityσ_dcwith the free current, i. e.

J_free=σ_dcE,

whereσ_dcis assumed a purely real constant that would be obtained under steady-state conditions at infinite time, i. e.σ_dcis independent of frequency [78, pp. 40-41].⁷ The calculation of the free currentJ_freevia Ohm’s law was already introduced by Maxwell [126]. Furthermore, Maxwell also ascribed the polarisation to the dielectric displacement [126]. On the other hand, Maxwell only mentions the proportionality between the free currentJ_freeand the applied field (‘electromotive force’) and, in this context, introduces the reciprocal of the specific resistivity (‘specific resistance’)ρas proportionality constant [126]. In the syntax of this work, assuming a homogeneous, isotropic material, Maxwell defines

J_free=ρ⁻¹E.

Maxwell did neither specify whether the specific resistivity should be a constant, inde-pendent on frequency, or even identical to the dc value [126]. This work adopts the exclusive separation and, consequently, the interpretation of the free currentJ_free by Jonscher [78]. Otherwise, the conversion into frequency domain would have also lead to an imaginary component of the free current, i. e. in quadrature with the driving field, and

7As in the derivation, the material is assumed homogeneous and isotropic. If not mentioned explicitly, the assumptions of the derivation are used also in this section.

frequency dependence (also of the derived components) resulting in a different electrical equivalent circuit. The derivation would have been more complicated, but still possible.

There is, however, another reason why the exclusive separation of dc and ac contributions is used although it is only an approximation: in the following part of the section, it will be shown that within the frequency range of dielectric relaxation, i. e. the range relevant for immittance spectroscopy, the assumption of the exclusive separation of dc and ac contributions is a valid approximation.

The above mentioned fact, that this exclusive separation is an approximation, becomes apparent when frequency dependence is introduced for the static transport process.

The frequency dependence becomes unavoidable, because the dependence on external parameters, that is included, can lead to an implicit frequency dependence. Generally, the resistivity might be dependent on the applied field, which is a quantity that is varied (periodically) in the experiment with varying frequency. A dc conductivity model, which usually assumes the system has reached its steady state (under static conditions), would independent of frequency lead to purely real values for the resistance, only. The response of a purely real-valued resistor, however, would bealways in phase with the driving field and result in theequilibrium current of the currently applied field at any time, independent of the frequency of the signal. In the following paragraphs it is assessed up to which frequency the approximation of utilising the dc model may be assumed valid. There are two very different types of conduction that are considered separately:

conduction through delocalised and localised states.

The situation is well investigated for the former case:metallic conduction, i. e. conduc-tion through delocalised states (e. g. confer [143, pp. 16-22]). In a basic descripconduc-tion, the equation of motion for a charged particle is assumed to consist of the constant accelera-tion due to the applied field and a fricaccelera-tional term that reduced the speed of the particle through collisions [143, pp. 16-22]. The mean-time between collisions is reciprocally linked to the dampening constant of the friction term. In a typical conductor it is of the order of10⁻¹⁴s, e. g. for copperτ=2.4·10⁻¹⁴s[143, pp. 16-22]. From such mean times between collisions and other material specific parameters the plasma frequency can be de-termined [143, pp. 16-22]. It is found, that the conductivity of a metal may be considered independent of frequency as long as it is sufficiently lower than the corresponding plasma frequency which is in the terahertz range for typical metals [143, pp. 16-22]. Hence, the form of electrical equivalent circuit proposed in this work can, for a metallic part of the system, be used in all typical frequency ranges of immittance spectroscopy.

As shown in this and the two following paragraph, in parts withconduction involving localised states, this separation remains valid up to frequencies sufficiently lower than those at which modulation quanta are absorbed or emitted by the charge. At a first glance one might think that thelifetimeof a trap, that is the time a charge carrier occupies a trap before it is released, might be connected to the high-frequency limit of the approximation in a material where charge transport is dominated by processes involving localised defects (for simplicity, but without the loss of generality, only one type of trap is assumed). Using the lifetime of the charge carrier in the trap (in combination with the corresponding trap density, energy differences, possible final states, recapture probability, etc.), a net jump rater_ifor charges successfully leaving the trap in a unit volume can be derived. As no

quantitative result shall be calculated, it is here not relevant what fraction of these jumps lead to an actual contribution to the current, as a result, without the loss of generality the net jump rater_ishould only include the fraction of those jumps that do contribute.

Generally, the jump rate may be influenced by external parameter, e. g. the applied field or the temperature. Consequently, it is implicitly dependent on time. It is not explicitly dependent on time, because the trap should not memorise prior conditions, the trap system shall be time invariant (the derivation was restricted to a ferroelectric, non-magnetic material). This means that the probability of leaving the trap has as such no time dependence, i. e. applying thesame conditionstoday, tomorrow any other day will result in the same probability. Of course, changing the conditions may change the probability, e. g. a higher temperature could lead to an increased jump probability. Using the familiar syntax of this work for parameter dependence while additionally stressing the fact that the parameters themselves are explicitly dependent on time the jump rate might be written as

r_i€

E(t˜ ), ˜P(t)Š

. To calculate the number of successful jumpsn_i in a time interval(t₁,t₂) may than be calculated by integrating the net jump rate

n_i= Z t2

t₁

r_i€E(t), ˜˜ P(t)Š

dt. (5.1)

For increasing frequency the length of the time interval, at which the condition can be assumed constant, decreases. However, the calculation of the total number of successful jumpsn_iremains valid, even when the length of the time interval falls below the mean lifetime (but is still above the already mentioned actual limit), because, as long as the number of traps is sufficiently high, the corresponding fraction of them still performs jumps, i. e. the number of successful jumps will decrease, but according to the ratio of the constant condition. For example, if the field is applied for only a tenth of the time to have Nsuccessful jumps, in the mean only₁₀^N of the jumps occur. Hence, interval length below the mean jump rate (but above the frequencies for absorption or emission of modulation quanta for the charge) result in a reduced current, but do not represent a general limit.

Another important time constant for conduction processes involving localised states is thetraversal time. This is the time the trapped charge requires actually passing through the barrier [16]. Usually this time is much shorter than the lifetime of an occupant in the trap [3]. In their work [16], Büttiker and Landauer discuss the influence of electrical fields varying with time on the barrier and the corresponding traversal time through it. They find that, as long as no modulation quanta is transferred, the barrier remains effectively the same. There are two cases: First, the applied signal has a low frequency, with a huge wavelength in comparison to the barrier width, in this case the situation for the charge looks like the respective static condition. Second, the applied signal has a wavelength in the order of or sufficiently higher than the barrier width, corresponding to a frequency in the order of or higher than the inverse of the traversal time. In latter case, the barrier becomes actually modulated, but the relevant effective barrier remains identical. As a result the frequency range of validity (of utilising a dc model while ignoring any delay for the resistance) is not dependent on the traversal time (confer [16]).

Finally, if the frequency reaches a range where modulation quanta are absorbed or

emitted by the charge carrier, the probability of passing through the barrier is changed [16]. At such frequencies the rate integrated in (5.1) deviates from the dc behaviour and the approximation becomes invalid. It is entirely possible that this frequency is lower than the inverse of the traversal time. The moment, modulation quanta become important, is not only when the proposed approximation breaks, but also the upper frequency limit of the range at which immittance spectroscopy is typically performed.

According to the definition by Jonscher [78, p. 6], dielectric relaxation is observed in the ‘“low-frequency” sub-quantum’ limit and, hence, ends below the above described frequency range of charges interacting with modulation quanta. As a consequence, for frequencies around or larger than this limit, the static conductivityσ_dchas to be replaced by a dynamic conductivity that recognises the frequency of the applied signal and allows a delayed response (an imaginary part that represents the component of the conductivity in quadrature to the driving field). In the limit of low frequencies the dynamic conductivity must converge against the static conductivity. At the above described high-frequency limit the question arises how to separate the response into dc and ac part. It might well be that such a separation is neither needed nor possible any longer. Also, the effect of dissipative tunnelling through the barrier was recognised in the work of Büttiker and Landauer [16]. While this would lead to longer traversal times, it does not change the above derivation of the frequency limit.

Regarding the influence of dissipation: the processes of overcoming a barrier by thermal emission, thermionic emission or even through tunnelling are usually dissipative, e. g. they involve absorption or emission of a phonon or photon. Even for tunnelling processes, because initial and final states may not be of an identical type and, hence, require the absorption or emission of a phonon or photon to conserve momentum or energy [37]. The statement of Jonscher that the frequency range of dielectric relaxation is in the ‘subquantum limit’ is of course not intended to restrict in this context.

This section does only assess the applicability of dc models to correctly account for their contribution to the free current densityJ_freein situations with ac signals. It does not asses whether they correctly contribute to the polarisation. Staying within the example of transport through randomly arranged localised states: The random arrangement of these traps leads to the formation of percolation paths through the material. With increasing frequency some jumps in these paths might become almost impossible to overcome. The charges trapped in a region of more easy jumps will then align their spatial distribution in the available traps with the applied field while other traps remain unfilled and, hence, represent a certain net charge. The movement of the remaining empty traps with the applied field can be interpreted as the alignment of bound charges with the applied field.

Furthermore, this process is, in almost all cases, lossy. With increasing frequency the regions of possible jumps become smaller and smaller, until reaching the limit of this polarisation process: the pair approximation. In this high-frequency limit a single mobile charge is aligning between only two traps. After this most elementary form of dipoles by traps cannot follow the applied field any longer, the contribution by hopping ceases.

In the presented assessment of the frequency limit of the actual dc conduction model, this conversion from free-current contribution to a dielectric contribution was ignored entirely. In fact, a part (probably even the major part) of the measured constant phase element is very likely due to this effect (compare the discussion about the value of the static permittivity in section 5.6.3). A static conduction model does not include this potentially important feature. In the reviews of Long [104] and Elliott [40] various models to describe the ac contribution of randomly arranged traps are discussed. A microscopic unified theory that can explain both, conductive and dielectric processes jointly, is however still missing and deemed one of the major challenges in solid state physics (see section 6.2.4).

5.5 The underappreciated method: Enhancing current-voltage analysis using voltage-dependent immittance

spectroscopy

5.5.1 Goals of conventional current-voltage analysis

Current-voltage characterisation is very common method of analysis which is applies to many very different material systems. A very prominent application is theidentification of charge-transport processes(i. e. understanding the underlying physics) and the subsequent determination of its relevant parameters, e. g. Schottky-barrier height, concentration and energetic distance of donor or acceptor levels from the mobility edge, other activation energies with the help of temperature dependence (confer [176, pp. 84-96, 254-270, 279-286, 402-405]). Of course many parameters, including the given examples, may potentially be determined independently using distinct, sometimes better suited, methods of measurement. Still, current-voltage analysis is a very common and accepted method which is very often used to extract parameters of the charge-transport process, especially in combination with temperature dependence.

5.5.2 Challenges in conventional current-voltage analysis

Only very rarely all pieces of interest can be studied separately. One notable, quite common, challenge is the agglomeration or depletion of charges around the interfaces, leading to unavoidable as well as significant contributions in the current-voltage relation.

There is no shortage of awareness of the presence of these barriers, but the magnitude of their contributions seems sometimes underestimated.

5.5.3 Immittance spectroscopy as logical consequence? Similarities between the different measurements.

In this section, it will be shown that voltage-(and temperature-) dependent immittance spectroscopy can improve the processes of identifying the charge-transport process as well as the subsequent parameter extraction. Voltage-dependent immittance spectroscopy contains the capacitive response of the system furtherto the full (see details in 5.5.7) information of the corresponding current-voltage analysis while it adds, except for the different but very similar measurement setup, no additional requirements with respect to the sample. The basic idea is that the capacitive information can be used to distinguish and separately extract different serial impedances. In consequence, the current-voltage curve (plus any other external parameter dependence) can be separately extracted for each serial piece. The knowledge of specific models for the different resistances or capacitances is not mandatory, but can improve the differentiation.

5.5.4 Solution by immittance spectroscopy

With the complementary frequency information, this new approach allows distinguishing different contributions to the measured current-voltage characteristic. A typical serial distortion is caused by additional resistances through space charges at the interfaces.

A very common problem, which is already manageable without additional frequency-dependent information, is to reduce the influence of contacts. This can, in usual current-voltage analysis, already be done in two ways:

First, by utilising selected materials at the contacts, often in combination with specific pretreatments of the surfaces prior to metallisation and possibly complicated subsequent annealing programs (e. g. confer [6] for an overview of Ohmic contacts for III-V com-pound semiconductors). While these processes are known for common materials and anyway necessary to develop devices, in science one is often confronted with materials for which such processes are not yet known. Developing low-Ohmic contacts for such latter cases may be a tedious and lengthy process.

Secondly, which is normal procedure, by performing a measurement using the four-point probe method [162, pp. 2-21]. In this way the contribution of the contacts can (in many cases) be removed, since the measurement of the voltage-drop is separated from the current measurement. With the goal on gaining knowledge about a material rather than developing a device, the four-terminal sensing can in most cases render the step of finding low-Ohmic contacts unnecessary.

Our approach offers a third method of separation by utilising the distinct capacitive bypasses of the corresponding pieces. This can be useful in the (admittedly) rare cases where a four-point probe method is not applicable and, more importantly as explained below, when the relevant piece of interest is itself a (hetero)structure consisting of several homogeneous pieces.

As distinguishes from the four-point probe method, the presented approach is not limited to eliminate the serial resistance contribution of contacts, but can be used to distinguish various serial pieces. Hence, the benefits of this approach emerge primarily for more complicated sample structures with several serial pieces. Unlike for the four-point probe method which eliminates the (parasitic) distribution, the presented approach allows separate extraction of all individual distributions. For example, if the investigated system is a more complicated structure of several layers. It might be desired to characterise the bulk contributions of each layer and its interfaces separately. Instead of synthesising different samples to study each interface and each bulk material separately, the pile of all layers can be measured together and different contributions extracted.

5.5.5 Requirements for and limits of the distinction of different pieces using immittance spectroscopy

A necessary prerequisite remaining is that at least at some point in parameter space the piece(s) of interest must have a significant contribution. The presented approach requires a measurable contribution of the piece(s) of interest in the total impedance of the system.

With decreasing impedance ratio, of the piece(s) of the sample that should be extracted

normalised on the total impedance of the system, the dynamic of the extracted data for the piece(s) of interest decreases. In addition to the increasing dynamic requirements, natural limits in the maximal applicable (total) bias voltage, usually because of some threshold at which the sample (or some piece of it) gets irreversibly damaged, might significantly reduce the possible voltage range for the piece(s) of interest, for a decreasing impedance ratio. Hence, for the explained method of extraction to work, the impedance of interest should not have little contribution as compared to other serial pieces of the system. Otherwise, the extraction of the current-voltage characteristic of the piece(s) of interest becomes increasingly difficult.

For example, if the piece of interest is the bulk of material X and the metal contacts at both sides form a depletion layer in X that is very high-Ohmic in comparison to the resistance of the bulk of X, this approach might not help to avoid developing a low-Ohmic contact on X. For positive and negative voltages, the depletion-layer width would be increased on one side and take the major part of the voltage drop. Further to whether the bulk resistance can be extracted at all, which depends on the dynamic of the measurement setup, the voltage range of the current-voltage curve would be limited to the maximal and minimal voltage drop of the corresponding part. Hence, for the given example, it might still be necessary to find a way of creating a low-Ohmic contact to the material X.

In summery, the piece(s) to be extracted needs to be a significant or, better, dominant contributor to the total impedance. In any other case, the parasitic contributions have to be reduced, e. g. , continuing in the context of the example given in the introduction of this subsection, better contacts or samples for the separate parts alone are imperative.

In summery, the piece(s) to be extracted needs to be a significant or, better, dominant contributor to the total impedance. In any other case, the parasitic contributions have to be reduced, e. g. , continuing in the context of the example given in the introduction of this subsection, better contacts or samples for the separate parts alone are imperative.