Impedance Spectroscopy 7

The advantage of an AC method is that there is nonet movement of ions, thereby eliminat- ing the need for an ion source. This method is implemented by placing an ionic conducting material under an alternating electric fieldE, with an angular frequency ofωand amplitude

E0, which can be described by the complex time (t) dependent wave function

The current response I(t) generated by this electric field in the material being tested, as depicted in Figure 2.1, can be described by a similar time dependent wave function with some applitude I0, plus some phase shiftφ,

I(t) =I0·ei(ωt+φ). (2.16)

Figure 2.1: Depiction of the real (<) component of an alternating applied electric fieldE(t) with an amplitudeE0 and angular frequencyω (solid line), and the real component of the induced current response I(t) of a material with an amplitude I0 and angular frequency ω phase shifted by some amountφ(dashed line).

From Ohm’s law,

E(t) =I(t)·Z, (2.17)

whereZ is the complex impedance characterized by a real componentZ0 and an imaginary component Z00,

Z=Z0+iZ00. (2.18)

It is convenient to define the reciprocal of impedance, or admittance Y, as

Y ≡ 1

Z =Y

0_+iY00_{. (2.19)}

Rewriting Ohm’s law using admittance

Y(φ) = I(t) E(t) = I0eı(ωt+φ) E0eıωt = I0 E0 (cosφ+isinφ). (2.20)

then equation 2.20 becomes

Y(0) = I0

E0

= 1

R, (2.21)

where R is taken to be the real resistance of the material under test. As the frequency

ω increases, the material’s current response due to mobile charge carriers begins to lag behind the applied electric field by some phase shiftφ. This, in turn, leads to a capacitive response from the material under test. This capacitive response is at a maximum at some characteristic frequencyω0, when the current response is exactly 90◦ out of phase with the

applied electric field, or when φ = π₂. Capacitance C, defined in terms of applied electric field and charge q,

C ≡ q(t)

E(t), (2.22)

can be used to evaluate the imaginary component of the admittance, by substiting in for

q(t) into the definition of current,

I(t)≡ d

dtq(t) =C d

dtE(t). (2.23)

Now, substituting in forE(t), using equation 2.15, gives

I(t) =iωCE(t). (2.24)

From Ohm’s law, equation 2.20, and the above result, the imaginary component of the admittance for φ= π₂ is Y π 2 = I(t) E(t) =iωC. (2.25)

With the real and imaginary components (at φ = 0, and π₂, respectively) of the complex admittance the complete polar form can be written:

Y =Y0+iY00= 1

R +iωC. (2.26)

Similiarly, with some rearranging the complex impedance can be written:

Z = 1/R

(1/R)2_{+ (ωC)}2 −i

ωC

(1/R)2_{+ (ωC)}2. (2.27)

rather than the admittance; however, the result is an expression in which the real component of the impedanceincreases with frequency, which is not phenomenological.

In Figure 2.2, the complex impedance as a function of frequency ω or “Nyquist” plot of equation 2.27 is presented. Here the apex is defined by a characteristic frequencyω0 in

terms of the resistive and capacitive response of the material under test,

ω0 =

1

RC, (2.28)

and the diameter of the semi-circle is given by the real resistance of the material R. These results lead naturally to a phenomenological equivalent RC circuit model, depicted in Fig- ure 2.3, which is commonly employed in the analysis of AC impedance results.

Figure 2.2: Nyquist plot (Z0 versus -iZ00) of the AC impedance of a material as a function of frequencyω. The diameter of the the semi-circle yields the real resistance Rof a material, whereas the apex occurs at a characteristic frequencyω0 equal to 1/RC.

Figure 2.3: A resistor (R) in parallel with a capacitor (C) circuit used to model the AC impedance response of a material.

Real material impedance responses rarely exhibit perfectRCequivalent circuit behavior. The most prominent deviation observed in real material impedance spectra is a depression of the observed semi-circle in a Nyquist plot, shown in Figure 2.4. Phenomenologically,

this behavior is accounted for by introducing a new circuit element Q, or constant phase element (CPE), in place of the capacitor. Where Qis defined as

Q≡(iω)nY0, (2.29)

wherenand Y0 are parameters characterizing elementQ. The admittance of anRQ circuit

is Y = 1 R +Q= 1 R + (iω) n_Y 0, (2.30)

where in the limit as ngoes to one, we get our previousRC circuit,

lim n→1 1 R + (iω) n_Y 0 = 1 R +iωC. (2.31)

Unfortunately, no good physical description has yet been given for element Q, nevertheless it is of considerable utility in least square refinements of real data sets.

Figure 2.4: Nyquist plots ofRC andRQ equivalent circuits, whereR is a resistor,C a capacitor, and Q(= (iω)n_Y

0) a constant phase element depicting the apex depression often observed in real AC impedance spectra.

The conductivity of materials was characterized by this method using an HP 4284A Precision LCR Meter, in the frequency range of 20 Hz to 1 MHz and an applied voltage of 1.0 V. Both single crystal samples and polycrystalline pellets were measured, with Ted Pella silver paint serving as the electrode material. Data were collected on samples upon heating and cooling in a tube furnace capable of temperatures up to 1200◦C under various atmospheres.

Least squares refinements using the previously described RQ equivalent circuit model were carried out using the commercially available software package, ZView (Scribner &

Associates). The refined material resistance R was then normalized by a geometric factor

A/l, where A is the area of the sample and l is the thickness of the sample to obtain the conductivityσ,

σ= 1/R

A/l. (2.32)

Then plotting the conductivity σ data as function of temperature T, in an Arrhenius plot (lnσT versus 1/T), the Arrhenius law:

σ(T) = A0 T exp ₋ ∆Ha kBT