Impedance spectroscopy

Impedance spectroscopy is used to characterise electronic components and to analyse various liquid and solid materials. A major application for solids is the investigation of transport properties of charge carriers. Through analysis of frequency domains, unwanted polarisation effects at the electrodes or other interfaces can be ruled out and low electrical conductivities can be measured.

The methods of impedance spectroscopy and of data analysis are as diverse as the applications of this measurement technique. In the following, only the subject relevant for this study is described.

An alternating voltage (AC) is applied to the sample via two electrodes, brought into contact with the sample by dipping them into an electrolytic liquid or attaching them to the surface of a solid. Solid samples are usually planparallel plates with the electrodes attached to the faces. The alternating voltage can be described by the equation:

U =U(t) =U₀sin(ωt+φ_U) (10.1) U0 is the amplitude,ω = 2πf is the angular frequency, andφU is the phase an-gle, which depends on the definition of the zero point of time. For convenience, the alternating signal is written in the complex form.

U =U(t) =U₀cos(ωt+φ_U) +isin(ωt+φ_U) (10.2) The voltage induces an alternating current in the sample, which is phase shifted with respect to the voltage and described by

I =I(t) =I₀cos(ωt+φ_I) +isin(ωt+φ_I) (10.3)

shift. The impedance analyser measures the amplitude and the phase shift of

Figure 10.1.: Principle of impedance spectroscopy: A sample of thickness d is placed between two electrodes with area A. The induced cur-rent is characterized by it’s amplitude and phase shift. The real part (I₁) of the complex electrical conductivity is proportional to the amplitude and the imaginary part (I2) is proportional to the phase shift. The picture was copied from Funke (2002).

the induced current. From the two quantities, the current signal can be divided into two parts, one oscillating in phase with the voltage, I₁(t), the other one is phase shifted by π/2, I₂(t) (Fig.10.1). The amplitudes of the two currents are proportional to the real and the imaginary part of the complex electrical conductivity, respectively.

σ⁰(ω) = I0,1

U₀ · d

A (10.4)

σ⁰⁰(ω) = I_0,2 U₀ · d

A (10.5)

σ(ω) = σ⁰(ω) +iσ⁰⁰(ω) (10.6)

d is the thickness of the sample, A the area of the electrodes, and σ(ω) the admittance, the AC-equivalent to the conductivity to continuous current (DC).

The inverse of the admittance is the impedance, the AC-equivalent to the DC-resistance. The applied voltage is small, between 1 and 1.5 V, to maintain the linear ohmic relationship between voltage and current. The real part of the electrical conductivity, σ⁰(ω), is a measure for the dissipated energy in the system and can be regarded as an analogue to an ohmic resistance. It is called the conductance and reflects the translational jumps of ionic and electronic charge carriers. The imaginary part, σ⁰⁰(ω), represents the stored energy in the system, analogous to a capacitor. It is also called the susceptance, as it is a measure of the ability of a material to interact with an electrical field in terms of rotational alignment of permanent dipoles and polarisation of atoms and molecules.

To investigate the transport properties and mobility of cations in orthopy-roxene, only the conductance needed to be considered. The electrical con-ductivity was deduced from the conductance spectrum, log(σ⁰) versus log(f);

the frequency f is given in Hertz, σ⁰ in Siemens per meter (Fig. 10.2). The conductance spectrum ideally consists of four regions: polarisation at low fre-quencies, the low frequency- or DC-plateau, the frequency dispersive region at higher frequencies, and the high frequency plateau at even higher frequen-cies. If no polarisation would occur at the interface between the sample and the electrodes, the DC-plateau would extend to zero frequency, corresponding to continuous voltage. The polarisation at the electrode-sample interface is expressed in a decrease of log(σ⁰) towards the low frequencies. This region extends the further with increasing temperature. The DC-plateau is the fre-quency domain of interest for this study. The level of the ”DC-conductance”

T1 DC−plateau

polarization

high−f−plateau

T1<T2

log(f [Hz])

T2

σlog( ’ [S/m])

Figure 10.2.: Schematic conductance spectra. f is the frequency, T is arbitrary temperature.

is the sum of the contributions of all mobile charge carriers to the electrical conduction.

σ =X

i

σ_i⁺ +σ_e⁻ (10.7)

where the sum is taken over all cations i⁺, e⁻ denotes electrons. The oxygen anions in silicates are very immobile due to their large atomic radius com-pared to the cations and are therefore assumed to not contribute to electrical conduction.

The frequency dispersive region and the high frequency plateau also arise from movement of the charge carriers, but the higher the frequency, the more thermal atomic oscillations around their equilibrium position contribute to the signal. If the frequency approaches the Debye frequency, i.e., the maxi-mum vibrational frequency of the crystal lattice at a given temperature, the spectrum again flattens out to the high frequency plateau, where all oscilla-tory movements of charged particles contribute to the conductance. The high frequency plateau was never observed in this study, because it appears very

rarely below 10⁶ Hz. The kinetic energy of the atoms increases with increasing temperature, and hence the frequency of successful jumps of charge carriers to an adjacent new equilibrium position increases. Therefore, the DC-plateau ex-tends to higher frequencies with increasing temperature. Due to the increasing mobility of charge carriers with temperature, the level of the DC-plateau then also increases, i.e., the whole impedance spectrum shifts towards the upper right.