##
1Electrochemical impedance spectra for the complete equivalent circuit of diffusion and reaction under steady-state recombination current
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Electrochemical impedance spectra for the complete equivalent circuit of diffusion and reaction under steady-state recombination current

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We have analyzed the impedance model for diffusion and reaction in non-equilibrium steady-state, when large diffusion current flows in a thin film either due to homogeneous reaction or heterogeneous charge-transfer in a nanoporous film (photoelectrochemical solar cells).

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Besides the standard diffusion resistance, transport in non-equilibrium conditions introduces a current generator in the transmission line equivalent circuit, which is due to an inhomogeneous conductivity.

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Numerical simulation of impedance spectra shows that the effect of the current generator is rather large when the diffusion length is shorter than the film thickness.

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## Introduction

5

The impedance spectroscopy technique is widely used in electrochemistry and solid-state electrochemistry, particularly for investigation of new and interesting systems such as nanostructured semiconductor electrodes and photoelectrochemical solar cells.

^{1,2}
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Diffusion and diffusion-reaction phenomena appear often in these systems.

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The impedance models are usually represented by a repetitive spatial arrangement (transmission line) involving resistances and capacitors.

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8

These elements pertain to the kinetic model that combines transport, reaction (or recombination) and charge accumulation processes.

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9

With simplifying assumptions these models can be solved completely and the impedance responses, depending on system parameters, have been classified.

^{1}
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Frequently, significant deviations from the basic diffusion-reaction model spectra are observed in impedance measurements.

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This is a current generator that occurs in the transmission line equivalent circuit when steady-state current flows in the transport-reaction zone.

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14

This element was derived by Sah in the transmission lines for solid-state electronics.

^{10–12}
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15

It has so far been neglected in solid-state electrochemistry, probably because the electrodes are usually blocking to steady-state current (

*e.g*., in intercalation electrodes and conducting polymer films).
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16

However, strong steady-state diffusion currents become a quite important and even dominant feature in many applications.

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17

For example, TiO

_{2}nanostructured electrodes in dye-sensitized solar cells^{13–15}provide strong diffusion-reaction (charge-transfer) currents at forward bias, and the same electrodes can be used in photocatalysis with a similar behaviour.
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18

In these conditions, the concentration is strongly non-uniform, which complicates the equivalent circuit model, so that analytical solutions are not generally possible.

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19

It is necessary to investigate the impedance response of these systems considering the full theoretical model of diffusion-reaction including the current generators identified by Sah.

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It is particularly important to evaluate the extent to which, and in which conditions, the effects of non-homogeneity and current generators modify the results of the simpler equivalent circuits

^{1}that are widely used.
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As a central example of electrochemical system in which diffusion and reaction are the dominant phenomena we will discuss the nanostructured semiconductor electrodes in contact with a redox electrolyte as shown in the schemes of Fig. 1.

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In this system electrons are injected either from photoexcited dye molecules adsorbed in the surface,

^{13–15}or from the conducting substrate in which the film is deposited, applying a negative bias potential with respect to the Pt counterelectrode.
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It is widely believed that the displacement of electrons is governed mainly by diffusion, due to the effective shielding of large-scale electrical field by the conducting medium that permeates the nano-pores.

^{16–18}
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In the case of substrate-injected electrons, the concentration will decrease towards the outer edge of the electrode (where the electron flux

*J*_{n}= 0) due to electron transfer across the oxide/electrolyte interface.
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There are two main situations according to the strength of charge transfer with respect to electron transport, as shown in Fig. 1d.

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For low recombination, the electron diffusion length is much larger than film thickness and the concentration profile is nearly homogeneous.

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In the opposite case, the concentration decays rapidly near the injecting boundary.

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Impedance models have been developed to treat these situations.

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The transmission line model used previously

^{1,2}is indicated in Fig. 1c.
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It corresponds to the expressionwhere

*R*_{t}is the diffusion resistance,*R*_{rec}is the recombination resistance,*ω*_{d}=*D*_{n}/*L*^{2}is the characteristic frequency of diffusion in a finite layer (*D*_{n}being the electron diffusion coefficient),*ω*_{k}is the rate constant for recombination,*ω*is the angular frequency and .
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The model of eqn. (1) is valid for homogeneous elements in the transmission line.

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In addition, eqn. (1) considers only the modulation of the gradient of the Fermi level

^{1}and not the current generators associated to position-dependent conductivity, as commented further below.
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With these simplifications, the transmission line model can be solved analytically.

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In the conditions of low recombination,

*R*_{t}≪*R*_{k}, eqn. (1) reduces to the expression
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35

On the other hand, in conditions of large recombination,

*R*_{t}≫*R*_{k}, the general impedance of eqn. (1) becomes the Gerischer impedance,
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Fig. 2 shows the impedance shapes corresponding to the transmission line model of Fig. 1c.

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The different shapes are obtained by changing only one parameter, the charge-transfer resistance

*R*_{rec}.
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Curves 1–5 are well described by eqn. (2), while curves 7–8 correspond to Gerischer impedance in eqn. (3).

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39

All impedances show at high frequency a diffusion line of slope 1, which is implied by eqn. (1),

*Z*(*ω*) ∝ (*i**ω*)^{−1/2}at*ω*≫*ω*_{k}.
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In the conditions of low recombination (

*ω*_{k}<*ω*_{d}) for curves 1–3, the 45° line of diffusion is a minor feature at high frequencies.
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The impedance is largely dominated by the reaction arc, the second term in eqn. (2), with the characteristic frequency

*ω*_{k}.
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In contrast, at high recombination

*ω*_{k}<*ω*_{d}, the impedance behaviour is similar to semi-infinite diffusion, curve .8^{1}
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The experimental impedance of nanostructured TiO

_{2}electrodes in solution, agrees qualitatively with the features of this model.
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Fig. 3a shows an instance of eqn. (2), and Fig. 3b shows the Gerischer impedance that is measured when the rate of charge-transfer becomes large.

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45

However, it is evident in Fig. 1d that in the case when the difusion length is shorter than the film thickness, the distribution of electrons is strongly inhomogeneous.

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This should have a considerable influence in the measured impedance, both through the inhomogeneous impedance elements and the current generators.

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In order to investigate this question, in this paper improved impedance models will be developed and discussed.

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We provide first a discussion of the physical origin of the complete equivalent circuit for diffusion-reaction of a single species.

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Thereafter we solve numerically the model in a variety of cases, varying both the materials and control parameters.

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We point out the main modifications to the standard model spectra.

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## The transport equations

51

We consider the statistics and transport of conduction band electrons.

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The number density

*n*is related to the electrochemical potential (or Fermi level)**_{n}by the Boltzmann distributionwhere*k*_{B}*T*is the thermal energy.
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According to non-equilibrium thermodynamics

^{19}one can formulate a linear phenomenological law that relates the irreversible flow to the thermodynamic force.
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For the flux of electrons in semiconductors the following relationship was formulated by Shockley

^{20}The equilibrium condition is .
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If we assume that the Galvani potential is uniform, as indicated in Fig. 1b by the homogeneous level of the lower edge of the conduction band, then

**_{n}=*μ*_{n}+ const, in terms of the chemical potential of electrons,*μ*_{n}.
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Eqn. (4) can be written as

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Then, by eqn. (5) the diffusion flux relates to the gradient of concentration as

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Eqn. (6) implies that the chemical capacitance

^{21}for electrons (per unit volume) takes the formwhere*e*is the positive electron charge.
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The meaning of eqn. (8) has been discussed elsewhere.

^{22–24}
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The equation of conservation for diffusion, recombination and generation iswhere

*τ*_{n}and*n*_{0}are constants for electron transfer (recombination), the lifetime and equilibrium concentration, respectively, and*Φ*_{ph}is the photogeneration rate.
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61

At

*x*= 0 we place a reversible contact to a conducting substrate (Fig. 1b) so that*V*, the bias electrical potential, relates to chemical potential of electrons in TiO_{2}as: −*eV*=*μ*_{n}−*μ*_{n0}.
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62

Using eqn. (6) we obtainAt the outer boundary of the film,

*x*=*L*, we have the reflecting boundary condition
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In the steady state, the excess electron concentration is determined by the equationwhich has been written using the electron diffusion length

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From eqns. (10), (11) and (12) the following distribution of electrons in the porous network is found for

*Φ*_{ph}= 0
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The diffusion current at the substrate/film interface gives the electrical current density

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## Small signal perturbation

66

The small signal perturbation techniques, such as electrochemical impedance spectroscopy, impose a small perturbation of a controlled physical quantity over a steady state.

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In general, this steady state will not be an equilibrium state.

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For example, when

*V*≠ 0 eqn. (14) implies ∂**/∂*x*≠ 0 at all*x*≠*L*, so that a steady-state current flows in the system.
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In order to analyze the evolution equations under a small perturbation, we express the different variables as a combination of a stationary and a (small) time dependent component: concentration,

*n*+*ρ*_{n}(*t*), flux,*J*_{n}−*i*_{n}(*t*)/*e*, Fermi level,**_{n}−*eφ*_{n}, and photogeneration rate,*Φ*_{ph}+*ϕ*_{ph}(*t*).
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70

We write the perturbation of both the flux,

*i*_{n}, and the electrochemical potential,*φ*_{n}, in units of electrical current and electrical potential, respectively, for convenience of representation in transmission line.
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We begin the analysis of small modulated quantities by the phenomenological relationship for the electron flux, eqn. (5), that describes the exchange of electrons between two neighbour points.

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When it is linearized, eqn. (5) giveswhere we have defined the transport resistance (per unit volume),

*r*_{t}, which is reciprocal to the electron conductivity
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The first term in eqn. (16) is the standard resistive component in the transport channel in transmission line models.

^{1,9}
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74

As a remark, we wish to clarify the meaning of the “potentials” in the transmission lines drawn below.

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Consider the difference Δ

*φ*_{n}between two points separated Δ*x*.
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The first term in eqn. (16) can be written aswhere Δ

*R*=*r*_{t}Δ*x*.
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Eqn. (18) is a generalized statement of Ohm’s law with respect to the local difference of

*electrochemical potentials*, Δ*φ*_{n}.
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Δ

*R*is the resistance in a slab of thickness Δ*x*.
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In the diffusion model (uniform Galvani potential) Δ

*φ*_{n}corresponds to a local difference in modulated component of chemical potential.
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If the steady-state concentration,

*n*, is uniform, the second term in eqn. (16) is zero and the small perturbation flux,*i*_{n}, will be driven by the difference of excess chemical potential, Δ*φ*_{n}, between two points.
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So the small perturbation component of diffusion flux is caused by two adjacent chemical capacitors,

*c*_{μ}, that will be differently charged, Δ*ρ*_{n}, by the difference in Δ*φ*_{n}.
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82

This is represented in equivalent circuit convention in Fig. 4a.

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This effect will also exist if

*n*is non-uniform, although the chemical capacitance will have different values at neighbor points.
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84

In addition, in this case the second term of eqn. (16) will not be zero, and another effect exists,

^{7,8,10–12}as discussed below.
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Assuming quasi-equilibrium, and linearizing eqn. (6), we get

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Then eqn. (16) can be written aswhere

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The second term in eqn. (20) was first identified by Sah.

^{10–12}
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Obviously, eqn. (21) is nonzero only if the system is not in equilibrium steady state, ∂

**_{n}/∂*x*≠ 0, (or alternatively,*J*_{n}≠ 0).^{7,8}
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This contribution is related to the small perturbation of the prefactor of the phenomenological flux expression, eqn. (5),

*i.e.*, the conductivity*σ*_{n}(through the carrier density, eqn. (17)).
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Indeed, when ∂

**_{n}/∂*x*≠ 0, a homogeneous change of the modulated component of Fermi level,*φ*_{n}, causes a change of the diffusion flux, which is represented by a current generator, −*g*_{σ}*φ*_{n}, in eqn. (20) and Fig. 4b.
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The conservation eqn. (9) governs the variation of carrier density at a given point.

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Eqn. (9) involves the time dependence through the term on the left.

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In particular, we consider a small sinusoidal perturbation of angular frequency,

*ω*.
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Linearizing eqn. (9), taking the Laplace transform d/d

*t*→i*ω*, and denoting the transformed variables by the same symbol,*i.e*.
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*φ*

_{n}(

*x,t*)→

*φ*

_{n}(

*x,ω*), we obtainwhere we have defined the recombination resistance (per unit volume)

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96

Eqn. (22) can be described as follows.

^{1,22}
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The first term is a capacitive current, the charging of the chemical capacitance.

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98

The second term is the recombination current.

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The third term is the photogeneration current.

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Note that these currents do not represent transport currents, but the rates of creation and destruction of conduction band electrons.

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101

Further discussion of these terms is given in .

^{ref. 22}
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102

The last term in eqn. (22) describes the change in concentration by transport current from neighbor points.

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103

Combining eqns. (20) and (22), the transmission line model of Fig. 5 is obtained.

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## Numerical results and discussion

104

With the aim of exploring the repercussion of the current generator on the overall response of a one-carrier system governed by diffusion and recombination/reaction with steady-state current different from zero, we have chosen a transmission line with completely reflecting boundary condition, as shown in Fig. 1c, in which the elementary line cell is that represented in Fig. 5.

Type: Experiment |
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105

For the sake of simplicity the photocurrent generator

*ϕ*_{ph}has been neglected,*i.e.*simulations were done under dark conditions.
Type: Experiment |
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ConceptID: Exp2

106

From eqn. (14) one can calculate the distribution of electrons for each value of the bias potential and thereby the circuit elements as a function of position.

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ConceptID: Met6

107

The chemical capacitance

*c*_{μ}results from eqn. (8), transport resistance*r*_{t}from eqn. (17), current generator element*g*_{σ}from eqn. (21), and finally reaction resistance from eqn. (23).
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108

The film thickness (0 ≤

*x*≤*L*) has been split into*m*slabs of equal width.
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ConceptID: Exp3

109

In order to check the resolution of the spatial discretization we compared the analytical value of the steady-state current [eqn.

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110

(15)] and the current resulting from the application of the Fick’s law to the first cell of the transmission line.

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111

Differences less than 0.1% are achieved for

*m*= 1000 for several values of the diffusion length (0.1*L*≤*L*_{n}≤ 10*L*) and the bias potential (0 V ≤*V*≤ 1 V).
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ConceptID: Obs1

112

It was also verified that the low-frequency limit of the impedance,

*R*_{dc}resistance, calculated by the derivative of eqn. (15) with respect of the bias potential, coincides with that obtained from the numerical simulation within the same accuracy.
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ConceptID: Res3

113

Fig. 6 shows the results of the numerical simulation of the exact equivalent circuit, as compared to the impedance spectra resulting under equilibrium conditions (zero bias potential).

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114

Simulations have been made for three different diffusion lengths, from much smaller to much larger than film thickness, and have been normalized to the dc resistance to facilitate comparison of spectral shapes.

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115

As commented before, the models without current generators display patterns which in the cases

*L*_{n}≥*L*consist of a transition from a purely diffusion impedance at high-enough frequencies (Warburg-like behavior) to a semicircle at lower ones, Fig. 6b and 6c.
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ConceptID: Obs3

116

For the diffusion length of order 3

*L*there is no modification of the impedance pattern because the concentration profile changes slightly with respect to the equilibrium situation.
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117

On the contrary, as

*L*_{n}decreases the effects of non-equilibrium and non-homogeneity become more visible in the impedance, Fig. 6a.
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ConceptID: Obs4

118

The high-frequency diffusional line of the Gerischer impedance becomes heavily distorted.

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ConceptID: Obs5

119

As previously commented, the bias potential has a great influence on the actual value of the dc resistance.

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120

However, the normalized impedance spectra only undergo changes for bias potential values within the range of 0–100 mV.

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121

For potentials larger than 100 mV normalized spectra exhibit patterns that coincide with that calculated for

*V*= 100 mV.
Type: Observation |
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ConceptID: Obs7

122

It is worth analyzing separately the repercussion of the current generator element

*g*_{σ}on the whole impedance spectra with respect to the influence of the inhomogeneity of the other circuit elements with position*x*.
Type: Object |
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ConceptID: Obj5

123

The effect of inhomogeneous circuit elements (without regarding current generators) was studied by others,

^{5,25}and it is exemplified in Fig. 7 in comparison with the complete equivalent circuit response.
Type: Result |
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ConceptID: Res6

124

As observed, the inclusion of the current generator is absolutely necessary for an exact determination of the frequency response.

Type: Conclusion |
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ConceptID: Con1

125

Not only the value of the dc resistance changes but also the shape of the impedance spectra is altered.

Type: Observation |
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Novelty: None |
ConceptID: Obs8

126

The potential change produced by the inclusion of the current generator can be calculated by considering the Thèvenin equivalent of the current source as

*g*_{σ}*r*_{t}d*x*, which using eqns. (17) and (21) may be written as a specific parameter given byIt is observed that*γ*is a dimensionless parameter.
Type: Method |
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Novelty: Old |
ConceptID: Met9

127

From eqn. (24) it is possible to define an average effect of the current generator for the whole film thickness by means of the following expression,

Type: Method |
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ConceptID: Met10

128

We point out that the average effect of the current generator depends on the concentration values at both limits of the diffusion layer and not on the particular concentration profile.

Type: Background |
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ConceptID: Bac14

129

In case of equilibrium conditions

*n*=*n*_{0}, constant along the film thickness, so that**= 0 as expected.
Type: Result |
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ConceptID: Res7

130

The effect increases as the ratio of concentrations at the boundaries becomes larger.

Type: Result |
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ConceptID: Res8

131

The variations of

*γ*with position for different diffusion length and bias potential are summarized in Fig. 8.
Type: Result |
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Novelty: None |
ConceptID: Res9

132

The effect of

*γ*is obviously higher as the diffusion length decreases.
Type: Result |
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Novelty: None |
ConceptID: Res9

133

The variations of

**with bias potential are displayed in Fig. 8e showing a saturation as the potential becomes larger.
Type: Observation |
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Novelty: None |
ConceptID: Obs9

134

In order to indicate the significance of the average

**defined by eqn. (25) we have calculated the increment of the dc resistance resulting from the transmission line calculations (without current generator) with respect to the exact simulation which considers*g*_{σ}.
Type: Method |
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Novelty: New |
ConceptID: Met11

135

By examining Fig. 9 one can realize that

**is actually a measure of the dc resistance deviation Δ*R*_{dc}.
Type: Result |
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ConceptID: Res10

136

Errors as high as 60% could be committed in case of neglecting the effect of the current generators on the calculated impedance.

Type: Conclusion |
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ConceptID: Con2

137

It is also of interest to check to what extent the commonly used impedance expression of eqn. (1) is able to extract the main parameters of the diffusion-recombination model.

Type: Goal |
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ConceptID: Goa5

138

It is known that eqn. (1) is equivalent to a transmission line with homogeneous elements (Fig. 1c) so that neither the effect of having an inhomogeneous distribution of circuit elements nor the repercussion of the current generators are taken into account in such a model.

Type: Model |
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ConceptID: Mod10

139

In Fig. 10, we show the variation of the diffusion coefficient resulting from fitting simulated spectra using eqn. (1): curves a and b for spectra which result from considering inhomogeneity only, and curves c and d for spectra generated using complete transmission line models, therefore including the current generator effect.

Type: Result |
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ConceptID: Res11

140

Although quite good fits were obtained in all cases, the diffusion coefficient is overestimated and this deviation is more apparent for lower values of the diffusion length.

Type: Conclusion |
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ConceptID: Con3

## Conclusions

141

We have analyzed the impedance model for diffusion and reaction in a spatially restricted layer.

Type: Object |
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ConceptID: Obj6

142

The complete equivalent circuit in conditions of inhomogeneous carrier distribution requires considering the position-dependent impedance elements, as well as the current generators in parallel to the transport resistances.

Type: Conclusion |
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ConceptID: Con4

143

As expected, these effects will be very small, with respect to the standard homogeneous transmission line equivalent circuit, when the diffusion length is much larger than the thickness of the diffusion zone.

Type: Result |
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ConceptID: Res12

144

In contrast, deviations of the impedance as large as 60% are found when the diffusion length is short.

Type: Result |
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Novelty: None |
ConceptID: Res13