The total current density in frequency domain

As immittance spectroscopy is most commonly performed and analysed in frequency domain [7, p. 4], the derivation will be continued in that domain. Since the time derivative

∂X˜(t)/∂ tof a functionX^˜in time domain becomesiωX(ω)in frequency domain, the Fourier transform of the total current density^˜J_totin equation (3.1b) is

J_tot(ω,E,P) =J^(stat)_free (E,P)+iωD(ω,E,P). (2.3) It should be noted, that the parameterωis not necessarily the frequency of the stimulus, just astis not necessarily the length of a period. Further, the term displacement current density in frequency domain shall only designate the termωD. As a result of this conven-tion, the real parts of the phasors of the displacement current density are the imaginary parts of the phasors of the total current density.

2.1.3.1 The displacement current in detail

As already explained in section 2.1.2.1, the electric displacement fieldDcan be divided in the two contributions from free spaceǫ₀Eand from the material, the polarisationP,

D(ω,E,P)=ǫ₀E+P(ω,E,P). (2.4) The polarisationPis, as also already mentioned in section 2.1.2.1, due to causality a quant-ity with complex phasors. It is a result of the (partial) alignment or induction of dipoles in the material. In frequency domain, it can be expressed by the product of the free-space componentǫ₀Ewith the dimensionless, frequency- and generally external-parameter-de-pendent electrical susceptibilityχ with complex phasors. Hence, the electrical susceptib-ilityχ is a measure of how strong the polarisationPin the material is in ‘units’ of the free-space componentǫ₀Eof the displacement current densityωD, i. e.

P(ω,E,P)=ǫ₀χ(ω,E,P)E. (2.5) Due to connection over the Fourier transform, in time domain the polarisationP^˜ is proportional to the convolution of the electric fieldEwith the electrical susceptibilityχ˜.

Complex and non-linear permittivity From Eq. (2.4) and Eq. (2.5) follows that the electric displacement field

D(ω,E,P)=ǫ₀[1+χ(ω,E,P)]E.

The term in braces is, conform to the usual definitions in standard literature (e. g. [78, pp. 36-47]) called relative permittivity and defined as

ǫ_r(ω,E,P)≡[1+χ(ω,E,P)].

As already explained, in absence of alternatives, dc current models are used to calculate the free current densityJ^⋆(stat)_free . These static models fail to take frequency dependence as

well as a delayed response into account. On the other hand, models for the susceptibility χ and all quantities derived from it (i. e. the [relative] permittivityǫ_[r], polarisationP and electric displacement fieldD) consider frequency dependence and delayed response.

As a consequence of the delayed response, which is, as explained below, equivalent to the fact that there can be no lossless polarisation process for anything but free space, the permittivity is a quantity with complex phasors. Separating the permittivity into itsreal component in quadrature with the applied field E (in compliance with the usual literature conventions, indicated by a ‘^′’) and theimaginarycomponent in phase with the applied field E (indicated by a ‘^′′’) leads to

ǫ(ω,E,P)=ǫ₀ǫ_r(ω,E,P)

=ǫ₀ǫ^′_r(ω,E,P)−iǫ₀ǫ^′′_r(ω,E,P)

=ǫ^′(ω,E,P)−iǫ^′′(ω,E,P), (2.6) where the quantityǫ, with its realǫ^′and imaginaryǫ^′′components, respectively, is the permittivity of the material [33, p. 94]. In the case of non-linear responses, the phasors of the permittivity are separated in their real and imaginary parts which exclusively belong toǫ^′ orǫ^′′, respectively. It should be noted that, due to the multiplication with the imaginary uniti, the real part of the permittivity contributes to the imaginary part of the total current density while the imaginary component of the permittivity contributes, in turn, to the real part of the total current density.

Following the notation of Debye [33, p. 94], K. Cole and R. Cole [27], Jonscher [78, p. 46], and others, as well as most engineering textbooks, the imaginary component of the permittivity is defined with negative sign. Both notations, negative or positive imaginary part, can be used equivalently. They result from the direction of rotation of the oscillating part of the electric field in the complex plane with increasing time. This work uses the notation with the oscillating partexp(+iωt)for the applied field, which means rotation of the oscillating part in mathematical positive sense, and results in the negative imaginary part (compare derivation of Debye in [33, p. 91]).

Illustratively, the polarisation response in a material due to an externally applied field cannot be instantaneous [164], because the (spatial) separation of charges or (at least partial) alignment of a dipole is always associated with some inertia and hence requires a certain, sometimes very short but still finite, time. A finite delay in time domain represents an unavoidable phase shift and, consequently, a non-vanishing imaginary part of the permittivity in frequency domain. Hence, for any material (i. e. everything but vacuum) the (relative) permittivityǫ_(r)is a quantity with complex phasors (confer [33, pp. 89-95]

for the linear case). Assuming causality, it can be derived from the Kramers-Kronig relations (which are also valid for non-linear systems [164]) that for any permittivity differing from the value of free space, the electrical susceptibilityχ of the material has to be, first, a function with complex phasors, second, its value must change with frequency and third,eachof its two componentsχ^′andχ^′′contains the complete information of the response [78, pp. 36-52]. As a result of their relation with the susceptibility, other frequency-domain representatives of quantities, namely the electric displacement fieldD, the polarisationP, the (relative) permittivityǫ_(r)and the displacement current density

ωD, have (except for free space) complex valued and frequency dependent phasors as well.

Since the real and imaginary parts of the phasors are exclusively associated with either g^′org^′′, with g=P,D,χ,ǫ_(r), the base function g is from now on called complex, if both parts g^′andg^′′are finite, independently of the form of the potentially complex oscillatory part. Furthermore, it is called purely real, if the part g^′′ is zero, or purely imaginary, if the partg^′is zero, although in the case on non-linear response, dependent on its form, the potentially complex oscillatory part of the harmonics may always force complex values.

Inserting the formulation for the complex permittivity, equation (2.6) in the total current density in frequency domain equation (2.3), results in:

J_tot(ω,E,P) =J^(stat)_free (E,P)+iω[ǫ₀E+P(ω,E,P)] (2.7a)

=J^(stat)_free (E,P)+iωǫ₀ǫ_r(ω,E,P)E

=J^(stat)_free (E,P)+iωǫ(ω,E,P)E (2.7b)

=J^(stat)_free (E,P)+ωǫ^′′(ω,E,P)E+iωǫ^′(ω,E,P)E. (2.7c) It can be seen that the imaginary part of the permittivityǫ^′′belongs to the component of the total current density in phase with the driving field and consequently contributes to the power loss [78, p. 45]. Thus,ǫ^′′(ω,E,P)/ǫ₀=ǫ^′′_r(ω,E,P)=χ^′′(ω,E,P)is often referred to as the dielectric loss [78, p. 45]. The real part of the permittivityǫ^′, on the other hand, is in quadrature with the driving field and, hence, in phase with the real part of the polarisation and electric displacement field [compare equation (2.8) with equations (2.3) and (2.7a)]. As a result, it does not contribute to the loss [78, p. 45].

Depending on the individual polarisation process, both parts of the complex permit-tivity can depend on various external parameters. As derived from the Kramers-Kronig relations, they are both always explicitly dependent on frequency [164]. Other dependen-cies vary with the individual polarisation mechanism: for example, in a material where the local dipoles are caused by field-assisted trapping and detrapping of carriers in an area of high defect concentration, analogue to the static transport [46], a dependence of temperature and electric field might be assumed. Another example, already mentioned above, may be the dependence of permittivity on applied pressure [31].

Usually, there is not one but various different polarisation processes with at very different resonance frequencies [78, pp. 50-51]. Around their resonances the loss becomes significant and both components of the permittivity change rapidly [78, pp. 50-51]. While the imaginary part of the permittivity has local maxima at the resonances the real part has smooth downward-steps with increasing frequency around each resonance while it is almost constant between resonances of large enough distance in frequency [78, pp. 50-51].

The smooth steps of the real part of the permittivity are downwards with increasing frequency since for frequencies much below the resonance, the polarisation process is fast enough to unrestrictedly contribute to the polarisation. At much higher frequencies the polarisation process is too slow to contribute. Around the resonance of the polarisation process the excitation of the process is most efficient leading to the highest dissipation of

Jtot

ωD_′ ωD_′′

Jfree

ω

E ℜ

ℑ

n=0.7 E

J_tot

J_free ωD

ωD^′′ =ℑ(Jtot) ωD^′

J_freeℜ(Jtot) log(J_x)

log(ω) Jtot

Figure2.3:Exemplary development of the total current densityJ_tot, free current densityJ_free, and the realωD^′and imaginaryωD^′′component of the displacement current density of a resistor bypassed by a parallel constant phase element withn≈0.7(leading to a constant phase of around

−63°) with increasing frequencyω. The value is comparable with the one extracted in this work for the constant phase element of the thin ta-C film. The electric fieldEis assumed to be purely real and the orientation of the vector is shown as a reference.

energy for the corresponding process. In equation (2.7c) it can be seen that, in the regions between the resonances, where the permittivity is almost constant and its imaginary part negligible, the displacement current densityωDwill be almost purely real and increase linearly with frequency. As shown later that is the reason for a capacitor to create a bypass linearly increasing in frequency between resonances.

2.1.3.2 Interpretation of the resulting total current density and its contributions

Including the formulation of the free current densityJ^(stat)_free based solely on static conduct-ivityσ_stat, equation (3.2a), in the formulation of the free current density, equation (2.7c), with inserted and explicitly written real and imaginary component of the permittivity, the final formula with geometry-less quantities for the total current density becomes

J_tot(ω,E,P) =

σ_stat(E,P)+ωǫ^′′(ω,E,P)+iωǫ^′(ω,E,P)

E. (2.8) In Figure 2.3 the development of the total current density and its different contributions is calculated for the common example of a resistor parallel to a constant phase element with n≈0.7which results in a constant phase of around63°. Similar to the reciprocal relation between impedance and admittance that changes the sign of the imaginary part, in the current density representation the phase of a capacitive element is, like in the admittance, positive and not negative, as it would be for the impedance.

Alternate interpretation: ac conductivity Instead of describing the current with the com-plex permittivityǫ, some communities define the complexac conductivityσ_ac, see e. g.

A

d A

E = E n, ˆ J

= J n ˆ σ

(E, P ) ǫ(ω, E, P )

µ_r =1

n ˆ

Figure2.4:Illustration of the exemplary geometry assumed in the derivation. The areaAshould be sufficiently large to ignore border effects. The material is supposed to be homogeneous, isotropic, non-magnetic and non-ferroelectric, hence, all field lines are assumed to be parallel and in the direction of the field which is applied normal to the surfaces with areaA. The length of the chunk of material isd, withd,A≪ ^λ(ω₂^max). The surfaces with areaAare normal to each other and their normal vector attacking on the centre or mass of the areas are on one straight line.

[40]. Using the assumptions in this work, the relation between these quantities can be described as

σ_ac^′(ω,E,P)≡ωǫ^′′(ω,E,P) (2.9a) σ_ac^′′(ω,E,P)≡ωǫ^′(ω,E,P) (2.9b) where real and imaginary part are reversed as compared to the complex permittivity [40]. One might think of the ac conductivity as expanding the free current densityJ_free, excludingthe dc contribution, with an additional real and imaginary term to describe the part of the displacement current densityωDof the total current density while completely omitting permittivity.