Impedance and dielectric spectroscopy revisited: Distinguishing localized relaxation from long-range conductivity

w

, _

> Pergamon

J. F’h~s. Chem. So/i& Vol. 55, No. 12. pp. 1491-1506, 1994 Copyright Q 1995 &evier Science Ltd Printed in Great Britain. All riphta reserved 0022-3697/94 57.00 + 0.00

IMPEDANCE AND DIELECTRIC SPECTROSCOPY REVISITED: DISTINGUISHING LOCALIZED RELAXATION FROM LONG-RANGE CONDUCTIVITY R. GERHARDT Georgia

Institute

of Technology,

School of Materials Science and Engineering, Atlanta. GA 30332-0245,U.S.A.

Abstract-The advantages of plotting a.c. data in terms of impedance,

electric modulus and dissipation factor simultaneously are illustrated. Complex impedance is generally employed for ionic conductors because it can easily distinguish between bulk and grain boundary effects. However, comparison with the modulus and dissipation factor data allows easier interpretation of the microscopic processes responsible for the measured a.c. response. In particular, the difference between localized (i.e. dielectric relaxation) and non-localized conduction (i.e. long range conductivity) processes within the bulk of the material may be discerned by the presence or the absence of a peak in the imaginary modulus versus frequency plot. Similarly, the absence or presence of a peak in the imaginary impedance versus frequency plot can be correlated to space charge effects and non-localized conductivity. Long-range conductivity results in nearly complete impedance semicircles but no frequency dispersion in the permittivity while localized conductivity is reflected in a frequency dependent permittivity but no measurable conductance. The degree to which these assignments may be made is related to the dielectric relaxation ratio (r =6,/t,) and the differences between the time constants of the different relaxation processes present in the material being examined Kqv~ords:

D. dielectric

properties,

D. electrical

properties,

1. INTRODUCTION

Since the early days of Cole and Cole [l], considerable progress has been made in utilizing complex plane plots and frequency explicit plots to explain the dielectric behavior and electrical conductivity of a wide range of solid state materials. Salient examples include the application of complex admittance to understanding ionic conductivity in yttria stabilized zirconia [2], utilizing the electric modulus formalism to understand dipole relaxations in ionically conducting glasses [3], Jonscher’s [4] ‘universality’ of the dielectric response and the development of equivalent circuits and fitting routines to help explain the possible mechanisms present [5]. One of the main advantages of frequency dependent measurements is that the contributions of the bulk material, the grain boundaries and electrode effects can easily be separated if the time constants are different enough [5] to allow separation. This of course requires data acquisition over a wide temperature range since the experimentally available range of frequencies is limited [6]. Because the dimensions involved in the grain boundary and electrode space charge phenomena are so much smaller than the bulk grains, it is often possible to isolate space charge effects and this has been done with great success in a number of systems [6-lo]. KS

55112-l

D. microstructure,

D. transport

properties.

One area in which further work is needed concerns the ability to determine if a bulk response is due to long range conductivity or to dipole relaxation. Both localized and delocalized conduction are bulk processes and will therefore give rise to the same geometric capacitance. Thus, an ambiguity exists in the interpretation of data, especially when it is not known whether the material is an insulating dielectric or a semiconducting or an ionic dielectric. The subject of the present paper is to demonstrate that plotting a.c. data in terms of impedance, electric modulus and dielectric permittivity simultaneously is extremely advantageous for distinguishing between these types of materials and in making the proper physical process assignments. First, the relationships between the different relaxation times [I I] will be presented in order to establish the mathematical basis for these assignments. Second, experimental data showing the difference between localized (i.e. defect relaxation) and non-localized conduction (i.e. ionic and electronic conductivity) processes within the bulk of the material will be demonstrated in three very different systems: (1) ionic conductivity and defect relaxation in trivalently doped cerium dioxide, (2) transport in porous silica with adsorbed moisture on alkali coated surfaces, and (3) electrical conductivity in semiconducting titania. The advantages of combining imaginary modulus with imaginary impedance

1491

R. GERHARDT

1492

plots to distinguish bulk and grain boundary effects were previously demonstrated by Hodge, Ingram and West [12]. More recently, attempts to utilize the modulus formalism to identify relaxation defects have also been made [13].

(a)

b

2. THEORETICAL CONSIDERATIONS The frequency dependent properties of a material can be described via the complex permittivity (c*) or dielectric constant (k*), complex impedance (Z*), complex admittance (Y*), complex electric modulus (M*) and dielectric loss or dissipation factor (tan a). They are in turn related to one another as follows: ~*=~‘_j~”

and

k* = c*/co

M* = M’ + jM” = l/c* Z*=Z’-jZ”= Y* = Y’+jY”

tan 6 = E”]~’ = M”JM’

(1) (2)

l/jcK,c*

(3)

=joC,c.* = Z’jZ”

(W

(4) = Y’/Y”

(5)

where w is the angular frequency w = 2zf, Co is the geometrical capacitance and j = ,/( - 1). In the case of a single relaxation process, a semicircle is obtained for all these functions when plotted in the complex plane. Whether a full, partial or no semicircle is observed depends on the strength of the relaxation and the experimentally available frequencies. In addition, these semicircles can take on slightly different shapes as described by various relaxation models known as the Debye relaxation [14], the Cole-Cole relaxation [1], the Davidson-Cole relaxation [15] and the Havriliak-Negami relaxation [16]. Mathematically the permittivity takes on the following forms:

Fig. 1. (a) Complex permittivity plot (6” vs E’) for the Lkbye relaxation model. (b) Real and imaginary parts of the pennittivity versus frequency.

plane plots, the data may also be plotted in terms of frequency explicit plots where either the real or the imaginary part constitute the y-axis. Thus, when the real part of the permittivity, c’, is plotted versus frequency, one may get a frequency independent straight line, a continuously decreasing curve or a sigmoidal shape curve depending on the strength of the relaxation, i.e. the ratio t&, If the experimentally available frequency range is wide enough to show more than a straight line in the real part, then the corresponding imaginary part, t”, will display a peak, t* = 6, + [(cs- c,)/(l + jwr)] (6) the shape of which will vary depending on which of the relaxation models apply. Figure l(a and b) depict c* = cm + [(ES- ~,)/(l + (joz)‘-a)] (7) a complex plane plot and frequency explicit plots for the real and imaginary parts in the Debye relaxation L* = 6m+ [(cS- ~,)/(l + joz)@] (8) model. Using the relationships given in eqns (l-5) and E* = cm + [(Es-c,)/(l + (jwr)‘-V] (9) eqns (6-9), the corresponding impedance, electric modulus, admittance or dielectric loss may be generrespectively where 6a is the real permittivity when OJ ated. Hence, all of the dielectric functions will result in semicircles when plotted in the complex plane and approaches zero and L, is the real permittivity when sigmoidal and peak curves for the real and imaginary o approaches infinity (i.e. when o is large for the parts of the function. As mentioned before, whether relaxation process under investigation), z is the perthese features are observable within the experimenmittivity relaxation time, a is the angle offset in the real imaginary plane and /I represents non-linearity in tally available frequencies depends on the strength of the relaxation and the value of the distribution the high frequency region. In addition to the complex

Impedance and dielectric spectroscopy revisited Table 1. Relationships between relaxation times of the different dielectric functions for the same relaxation process Cole-Cole .r

Debye model

model

ri

T

*

parameters. It is therefore not surprising that certain dielectric functions are favored depending on whether the material being measured is insulating, semiconducting or conducting. The electrical response of ionically conducting materials is generally reported in terms of impedance and/or electric modulus because long range conductivity dominates while that of more dielectric materials is given in terms of permittivity and loss because localized relaxation dominates. The difficulty arises when one wishes to establish when localized relaxation ceases and long range conductivity ensues. This is not easy to establish because both of these processes are considered as bulk behaviors and thus are proportional to the same experimental measurable quantity, the geometric capacitance. Before discussing how one might be able to distinguish these two physical processes, it is necessary to discuss particular

the meaning relaxation

of a relaxation

time for a

process.

It is often assumed in the literature

that a particular

1493

peak resulting from the SAME physical process for four different relaxation ratios (r = E~/E~) respectively. It can be seen that the electric modulus will show up at the highest frequencies while the imaginary part of the permittivity will show up at the lowest frequencies since t, 2 rr > 7tand> tZ > zy. The reason why a peak in e c is generally never detected at the same time as a peak in MM at the same temperature is because the commonly used frequency ranges are not wide enough to display a peak in both of them at the same time. Also as indicated, the relaxation ratio will determine how close or far apart the peaks will be.. The order of appearance suggested by Table 1 will be maintained only in the absence of other processes or when the relaxation times of other physical processes are very different from one another. An example of the interaction between two relaxation processes has been demonstrated by Hodge et al. [12] where the order and magnitude of the electric modulus and the imaginary impedance change based on the time constants of the two different processes. The distribution parameters CI and fi of the Cole-Cole, Davidson-Cole and Havriliak-Negami type relaxations will also affect the position of a relaxation peak on the frequency spectrum. The relationships for the Cole-Cole relaxation were also given in Table 1. As can be seen, the relaxation times maintain their original form as given for the Debye model, but are modified according to their distribution parameters. So far the discussion has focused on the dielectric parameters as they are normally given for dielectric materials. However, it is well accepted that long range conductivity may be represented in terms of a parallel RC circuit so that its admittance may be written as follows:

physical process has a single relaxation time, irrespec-

tive of what function is chosen to display it. This is not the case as each dielectric function has a relaxation time of its own, as demonstrated in an earlier paper [ll]. What this means is that a particular physical process can be represented by several relaxation times, depending on which dielectric function one chooses to use. Table 1 lists the relationships between the different dielectric relaxation times for the SAME physical process in the Debye and Cole-Cole relaxation models [ 111.As can be seen, the ratio r = es/t,, plays a very important role in determining the value of the relaxation time for a particular dielectric function and therefore it determines at what frequency the imaginary part of a particular dielectric function will have a peak (since wr = 1). Figure 2(aad) show schematically the position at which the different dielectric functions will display a

Y* = G(l + jmr)

(10)

where 7 = RC is the conductivity relaxation time [ 121. Utilizing the relationships given in eqns (l-5) leads to e* =c,(l

-jar)

Z* = R/(1 +jwt)

(11) (1-a

tan 6 = l/w7

(13)

M* = 1/[6,(1 - j/wr)].

(14)

Examination of these equations leads to the conclusion that in this instance only t,+, and 7= have a finite value and are exactly equal or very similar in value depending on the distribution parameter [l l] while 7<, ttand and 7y do not exist [ll]. It should be

R. GERHARDT

1494

noted that if the conditions of z, -+ co and cS+ co are applied to eqn (6), eqn (11) will be obtained thus providing the crucial link between localized relaxation and long range conductivity. Figure 3(a and b) depicts the difference between these two types of transport. Notice that even a relaxation ratio of 200 is sufficient to have them overlap when the response is purely long range.

3. EXPERIMENTAL EVIDENCE In order to illustrate the claims made in Section 2, three very different types of materials will be analysed. They are: (1) cerium dioxide doped with trivalent cations, (2) alkali surface doping of porous silica, and (3) electrical conductivity of titania. In all cases, some background data will be given for each material first so as to provide sufficient information to demonstrate the benefits of plotting the data using all of the dielectric functions simultaneously.

3.1. Cerium dioxide doped with trivalent cations Cerium oxide has been of interest because it possesses the fluorite structure and thus it is an ionic conductor even in the undoped state [17]. Doping it with aliovalent impurities has been shown [18-201 to result in considerably increased bulk ionic conductivity over its undoped state and that of yttria (or other) stabilized zirconia, which is used in oxygen sensors [21] and solid oxide fuel cells [22]. Nowick and co-workers [6,7, 18-20,23-40] and other investigators [4146] have published a large number of articles dealing with the effect of the type, size [20,37,38] and concentration of the dopant on the ionic conductivity [24,41], grain boundary effect [&S, 34, 35,44,45], dielectric relaxation [39,40,46], anelastic relaxation [36-391 and electrode effects [30-331 of these CeO, solid electrolytes. It has generally been accepted that ionic conductivity measurements are carried out at high temperatures (T > 150°C) and dielectric or anelastic measurements are carried out at lower temperatures (T < 25°C).

1.2 1

8,:

Tan 6

..

B

J

0.6 .-

E

0.4 .-

2

0.2 --

Log Frequency

Log Frequency

0

0

Log Frequency

Log Frequency

Fig. 2. Plots of calculated E“, YI’,tan 6, Z u and M ” versus frequency for the SAME relaxation process. Notice that frequency at which a peak will appear depends on the relaxation ratio r = cJL,: (a) r = 3, (b) r = 40, (c) r = 200 and (d) r = 10,000.

Impedance and dielectric spectroscopy revisited

w

(a)

12T\

oi

Log Frequency

(bJ

PM0

_

0:

Log Fnqmw Fig. 3. Localized relaxation results in separate 2 ” and M " peaks (small r) (a) while long range conductivity results in

overlapping 2 II and M * peaks (large r) (b).

1495

The higher temperatures are carefully chosen based on where a complete or nearly complete impedance semicircle can be obtained to detect the physical process one wishes to characterize. For relaxation measurements, lower temperatures are chosen because below room temperature the d.c. ionic conductivity contribution is negligible and one can detect the a.c. relaxations undisturbed. However, if a.c. measurements are carried out in the intermediate temperature range of these types of measurements, one can also distinguish localized defect relaxation from long-range conductivity. It has already been established that excluding electrode effects, the a.c. response of cerium dioxide doped with trivalent cations results in 2 semicircles in the impedance plane [6,7]. One of the semicircles represents the bulk behavior and the other grain boundary effects. These assignments were based on the geometric capacitance and show fairly good agreement with actual microstructures in ceria [7,35,45] as well as in zirconia solid electrolytes [8.47]. Figure 4 depicts two examples of this type of response: one at 375°C for a sample containing 6% LazO, and the other at 456°C for a sample containing 1% Sc,O, [7]. If many of these measurements are carried out over a temperature range, two different conductivity plots such as the ones shown in Fig. 5

t” (ohms)

-

(O’

ccoz:

375V

6% Lagos

(b)

456.C

(ohms) CeOi! : 1% Sc203

01

I

v

I

0.5

UI

I

I

I

I

1. 2’ (ohms)

Fig. 4. Complex impedance plots for ceria doped with 6% La20, at 375°C (a) and doped with 1% Sc,O, at 456°C which display the presence of a bulk and a grain boundary behavior [7].

R. GERHARDT

1496

are obtained [24], one due to the bulk behavior and the other due to the grain boundary behavior. Notice that 4-probe d.c. measurements agree with the lower curve because the interconnected grain boundary phase dominates [35]. However, within the bulk of the material there are two a.c. conduction mechanisms: (1) long range a.c. conductivity, and (2) localized transport of oxygen vacancies around a trivalent impurity. The first conduction mechanism deals with long range motion of the vacancies introduced by the aliovalent doping [23] by way of charge compensation and follows the reaction CeO, + M,09 + 2M& + Vi where ML represents the trivalent impurity in a cerium site and Vi the oxygen vacancy and would result in a conductivity equivalent to the 4-probe d.c. conductivity in the absence of grain boundary effects. At low enough temperatures, association of the introduced defects occurs and gives rise to dielectric loss from the motion of a bound vacancy among all its equivalent positions [48,36-40] in a bound (M,V,)’ pair. The magnitude of the conductivity for the second mechanism is so small in comparison to that provided by the first that even though it is built

into the response it is often ignored. Wang and Nowick [40] did an extensive study where the effect of having just as many (M,,V,)’ pairs as unassociated M& in a series of yttria doped ceria samples were considered and related to the a.c. conductivity. Their interpretation was predicated on several types of measurements including thermal depolarization measurements. Sarkar and Nicholson [ 13,451, on the other hand, made the first attempts to derive information about both the long range conductivity of free vacancies and localized conductivity of bound vacancies in lanthanum and yttrium doped ceria using the modulus formalism. More often, the two mechanisms are dealt with separately as if they belonged to two different materials since long range conduction is reported at higher temperatures while dielectric loss is reported at lower temperatures. A case in point is that of scandia doped ceria whose long range conductivity was first reported in 1981 [20] and dielectric and anelastic relaxation were reported several years later [37-391. This is not surprising in view of the fact that the conductivity is so low for this material (and impedance is so high) that the impedance plot shows all frequency data practically parallel to the imaginary impedance axes at room temperature (see Fig. 6(a)).

1000 / T Fig. 5. Plot of conductivity vs reciprocal temperature for 1% Y,O, doped CeO, displaying the bulk and grain boundary conductivities calculated from semicircles similar to those depicted in Fig. 4 [6].

Impedance and dielectric spectroscopy revisited However, when the same data is plotted in the permittivity and the modulus planes (Fig. 6(b and c)), it becomes clear that more information can be derived from the room temperature frequency dependent measurements. Complex permittivity and complex modulus plots are particularly useful for extracting E, and E, from the plots since M’(co) = l/c, and M’(0) = l/c, [l I]. To get back to the subject of this paper, we need to look at the frequency explicit plots. In fact, the same data converted to tan 6 shows a complete dielectric loss (tan 6) peak in the frequency plane (Fig. 7(a)) which is identical to the dielectric loss peak obtained by making loss measurements at 1 kHz as a function of temperature [39]. Furthermore, when we compare tan 6 with M “, Y” and 2 ” (see Fig. 7(b, c and d)) we find that no peaks in Y fl and Z” exist and tan 6 and M” nearly overlap as would be predicted for a material with a small relaxation ratio (r = es/t,” < 9) [l I] since 7Un6= Jr?,” (see Fig. 3(a)). The small relaxation ratio is manifested in two plateaus visible in the real part of the dielectric constant versus frequency plot depicted in Fig. 8. In contrast to the scandia doped ceria data, the dielectric constant versus frequency plot for ceria

1491

doped with 1% Gd,O, shown in Fig. 9 displays only one plateau and an increasing k’ value toward the lower frequencies signifying a much larger relaxation ratio. Thus its tan 6 and M” plots at the same temperature are much further apart as demonstrated in Fig. lO(a and b) for 75°C and 125”C, respectively. In this case, we find that peaks in Y fl and Z fl do exist (since r > 9) and a peak in Z ” fits nicely between the tan 6 and M ” peaks (also shown in Fig. 10). The Z ” and M” peaks do not overlap in this case as this sample has components from both a long range and a localized relaxation. This is supported by the complex impedance plot depicted in Fig. 11 which shows a complete semicircle unlike the scandium doped case where only a portion was visible (Fig. 6(a)). It should be noted that the tail at low frequencies is possibly due to a grain boundary effect or the onset of a purely resistive conductivity. 3.2. Bulk porous silica with alkali coated pore surfaces Bulk amorphous silica is one of the most studied insulating materials because of its excellent dielectric properties [49]. SiO, is the native oxide of Si and as such it plays a very important role in the fabrication (b)

(a)

7------

1.6E6-.

6

0

0

;

: I ; I : I : 4E5 8E5 1.2E6 1.6E6

35

Z' (ohms)

37

39

41

43

k

Fig. 6. (a) Complex impedance, (b) complex dielectric constant, and (c) complex modulus plots for CeO,: 1% !k,O, measured at 25°C.

R. GERHARDT

1498

(a) 0.06 . 1

0.04..

s

.

.

.

.

.* ..

. . .

. .

. ‘.

0.02

.

t

t

-.-*w

I

lE7

IEB

.. 1ES

IIS

o.ooo;j4 A“..;;,

Frequency(Hz)

,

lE5

IES

ie

Frequency(Hz)

(d)

1.6E6

.

t 1.2E6 I

4E5

I -----* ?E4

.

. .**’

IES

0..

IEO

lE7

I)E4

IEO

.

I

I

I

.

Fmquency(Hz)

’ . .. lE7

lE6

lE6

1

.e

Frequency(Hz)

Fig. 7. Frequency explicit plots from the same data reported in Fig. 6: (a) tan 6, (b) M ‘I,(c) Y” and (d) Z “.

of integrated circuits for microelectronics [50]. In addition to thermal oxidation of Si to obtain SiO,, many other methods are often used to produce it. They include sputtering, electron-beam evaporation, sol-gel processing, PECVD and anodization with HF. Even though the silica surface is hydrophobic in nature [Sl], the presence of even minute concentrations of

46

unpaired hydroxyl radicals and/or other hydrophyllic species results in considerably enhanced ionic transport [52-541, thus interfering with its insulating function. However, this affinity to adsorb water renders it a good candidate for humidity sensing [5-O]. It is believed that porosity augments silica’s ability to adsorb water as a result of the larger surface areas

I

30-1

lE2

lE3

lE4

lE5

lE6

lE7

Frequency(Hz) Fig. 8. Real part of the dielectric constant vs frequency curve for CeO,: 1% Sc,O, sample at 25°C showing two plateaus indicating a small relaxation ratio.

Impedance

and dielectric spectroscopy

revisited

1499

150

. 120

. . 90

. .

4

lEl1El

.

lE2 lE3 lE4 Frequency (Hz)

.

1EE

.

60

l

.

. l

.

‘0.

! 30 t 0

lE3

lE2

lE5

lE4 Frequency (Hz)

Fig. 9. Real part of the dielectric constant vs frequency curve for CeO,: 1% Gd,O, sample measured at 75°C. Data displays only one plateau signifying a much larger relaxation ratio than the sample in Fig. 8.

(a)

-

I--

-M

tan delta II

0.6..

0.6..

0.2.-

O* IEI

lE2

(b)

lE3 Frequency (Hz) 125 OC

A

P--Yf--T

1 15

lE4

-

tandeita

-

M”

I -2”

0.6..

lE4

lE5

Frequency (Hz) Fig. IO.Clear display of the presence of same relaxation process using normalized tan 6,Z ” and M n plots vs frequency at two temperatures for CeO,: 1% Gd,O,: (a) 75”C, and (b) 125°C. Notice that all functions shift to higher fquencies as the temperature is raised.

R. GERHARDT

1500 5E6

Porous silica samples analysed below were prepared by a colloidal processing method [64] which has been extensively studied by the present author and associates [St-60,655731. The pore microstructure [66-69] and sinterability of the starting precursor material (15% dense) to full density [70-721 have been explored at length. As a result, dielectric property measurements as a function of density have been carried out in a dry environment as well as in the presence of water vapor [73,54]. The effect of surface doping with alkali ions to enhance water adsorption was done as early as 1987 [57] while varying the type and concentration of alkali ions in order to tailor the response has been done more recently [58-60]. Figure 12 presents data for a K-coated porous silica sample exposed to various relative humidities at room temperature [60]. This behavior shows that the peak in M” moves to higher frequencies as the relative humidity is increased. Similar plots are obtained for uncoated samples as a function of relative humidity [54]. Figure 13 shows the impedance plot for an uncoated sample [54]. As can be seen, the impedance of the samples decreases so rapidly with increasing humidity that it necessitates multiplication factors 7 x , 70 x and 400 x in order to display them all in one single graph. The impedance for samples exposed to even lower relative humidities than shown is so large that a complete semicircle is rarely observed. Taking the reciprocal of the intercept of these semicircles with the real axes and multiplying by the sample’s geometric factors will give a sigmoidalshaped conductivity curve. The plots of conductivity versus relative humidity for three different samples

125%

a

E 3E6 k 2E6 i

or : lE6: : 2E6; : 3E6: : 4E6: : 0

5E6

2’ (ohms)

Fig. 11.Complex impedance plot for same CeO,: 1% Gd,O, sample reported in Figs 9 and 10. [61]. In addition, processing methods used to introduce porosity often leave unbound OH- radicals further enhancing water adsorption. More recently, it has been shown that coating the pore surfaces with alkali ions allow tailoring of the response to different relative humidity environments [5760]. The mechanism by which these changes occur is still under debate, although there seems to be agreement that it takes the form of a Grotthus mechanism 1621. The increased ionic conductivity as a result of the adsorbed water has been demonstrated to be easily detectable by impedance measurements in the radio frequency range [54] even though water molecule relaxation does not occur until the GHz range [63].

I 0.06.-

. .

l

. ;

0.03.. .* m 0 lE2

.**

lE3

.

.

. . .

. .

.

* . .

St . d P AA AA lE4

= t

l *\

. i

* i

: A9

:, 4.

l

88. WI@ mm lE5

Frequency

A d-

. .

lE6

lE7

1 i8

(Hz)

Fig. 12. Imaginary modulus vs frequency plot for a K-coated porous silica sample measured at room temperature. Data for several relative humidities (RI-I) is shown. Notice that the peak shifts to higher frequencies as the RH is increased.

Impedance and dielectric spectroscopy revisited

0

300

600

1501

1 DO

900

2’ (M ohm) Fig. 13. Complex impedance plots for an uncoated porous silica sample at several relative humidities. Notice that data at the higher relative humidities has been multiplied by the factor indicated in order to plot them all on the same graph. are shown in Fig. 14. These curves are representative of what is normally observed for silica processed as

described. There are several points to note, however. One is that the increase in conductivity from low to high humidity can be as much as five orders of magnitude. The second item to note is that the conductivity of all samples increases slowly until a threshold humidity has been achieved at which point most of the increase ensues. As indicated in Fig. 14, the humidity at which the largest increase occurs depends on the type of alkali used [60]. Other factors that influence it are the precursor microstructure, the

concentration of the alkali solution used to coat the pore surfaces and the measurement conditions [59]. In all cases, a plateau or saturation point appears to be reached. It should be clear from the above discussion that at least two mechanisms must be operative in the measured behavior of porous silica as a function of humidity. It seems reasonable to assume that at low humidities, the measured response would be due to isolated pockets of adsorbed water while that at high humidities must be representative of long range conduction. In order to ponder these questions, we need to look at the imaginary functions: M “, Z M

lE+lE-5-IE-6--

0

20

40

60

80

% R.H. Fig. 14. Log conductivity vs relative humidity plot for porous silica samples coated with Li, Na and K salts of the same concentration. Each point was obtained from the corresponding complex impedance plots similar to the one depicted in Fig. 13.

R. GERHARDT

1502

0.2

(

.., 0.0 lE2

. .. . .

‘\

.

. .

.

:

: .

.

.:q

.

.

.

.

l

.

:

.

ti IFi3

lE6

lE5

lE4

lE7

1E6

lE9

Frequency (Hz) Fig. 15. Well separated peaks of normalized M “, tan 6 and Z 0 plots vs frequency for a K-coated porous silica sample measured at 39% RH.

terpretation will appear

and tan 6. Figure 15 displays three well formed peaks for a sample

doped

a 2N solution of KC1

with

measured at 39% RH. As would be expected from the conductivity plot, the dielectric response is essentially localized. In contrast, the behavior of the same sample measured at 66% RH (Fig. 16) shows an overlap of M. and Z . with no measurable value of tan 6 at those frequencies. It should be noted that the rapidly increasing value of tan 6 at lower frequencies is a manifestation of the onset of d.c. conductivity. These assumptions are validated by the frequency dependent permittivity or dielectric constant and complex impedance plots (not shown). Further in-

3.3. Undoped titania TiO, is an important material because of its ability to be fully insulating when stoichiometric as well as semiconducting at low PO, [74]. As a result titania is used as a dielectric as well as a gas sensor [75]. Because of the wide variation in electrical response, this is an excellent material to illustrate both insulating behavior and electronic conductivity. In order to do that, data taken at 25°C and 500°C from the same sample will be presented.

66% RH 1 .o--

l

.

.

iz

.

0.4--

. l

lE2

.

.

. ‘8

0.2..

o.o+

. .

.

1

z”

. . .

0.6..

.Y

. . .

.

.

F

2N K

. .

0.6..

of these plots and those of other samples elsewhere.

. ..lE3

I .. .. . . :, .’ . .* . .. . . lE4

. .. .

’

.

.

.

A

. .

. .

n

. . . . AA . . &_ . 1 0.. 1..b, ‘” lE7 lE6 lE5

1EI6

Frequency (Hz) Fig. 16. Overlapping Mu and Z * vs frequency peaks with an accompanying fast rising tan 6 indicating that the process being measured is of a long range nature.

Impedance and dielectric spectroscopy revisited

6000~

.

I

Iy

2000

lE2

5oooc

-

25OC

. .

. i

t 0

-

. .

t

1503

. .

.-

.

.:trrr

lE3

8 a U88W

.

.

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lE4

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Frequency (Hz)

Fig. 17. Dielectric constant vs frequency for undoped TiO, at 25°C and 500°C. Notice the absence of a frequency dispersion at 25°C but a large dispersion at 500°C.

Figure 17 depicts the dielectric constant versus frequency plot at the two temperatures. As expected, the behavior at room temperature is fully insulating displaying virtuaily no frequency dispersion. In contrast, the data at 500°C displays a large frequency dispersion never reaching the low frequency plateau. The same sets of data plotted in the complex impedance plane are depicted in Fig. 18. It can be seen that the insulating behavior at room temperature is so strong that the 2” values practically sit on the y-axis. The complex impedance data at 500°C on the other hand, results in a well formed semicircle accompanied by a tail representative of mostly resistive behavior. These results are consistent with the reported semiconducting behavior of TiO, at 500°C [751. The frequency explicit plots for the measured response of TiOz at 25°C and 500°C are given in Figs 19 and 20, respectively. At 25°C no peaks can be detected in either of tan 6, Z” or M” because current dissipation is minimal. Notice that they all appear close together as is expected when r is very small. At 500°C the characteristic of overlapping Z” and M” peaks for a long range conductivity process appears. This is accompanied by a fastly rising tan 6 at low frequencies which is representative of the presence of dc. conductivity. This behavior is similar to what was observed in Fig. 16 for porous silica at high relative humidities. Besides the overlapping Z” and M” peaks, an additional peak appears at lower frequencies in the Z” snectrum only for the TiOz sample measured at 500°C (see Fig. 20). The presence of this peak is a result of the nearly pure resistive behavior

seen in the complex impedance

plot

Fig. 18 and again provides evidence that frequency explicit plots provide detail not normally discernable. of

4. CONCLUDING REMARKS

Data for three very different materials with different types of physical processes were presented to illustrate the benefits of plotting frequency dependent measurements using not only Z ’ vs Z ’ plots but also, M U vs M ’ and E” vs 6’ as well as the frequency explicit plots, namely tan 6 vsf, M” vsf, Z” vsfand Y S vs f: The results analysed included ionic conductivity and defect relaxation in doped ceria, relaxation

7000

t

1.

l

25*C

6660

0

2E5

4E5

BE5

r

0

:,:

. . ..w-0 ... . . : : : : : : : i : i

1OCO206030604W6500666607060

z’ (ohms) Fi g. 18. Complex impedance plots for TiOz sample reported m Fig. 17. The sample is extremely insulating at 25°C but shows a small semiconducting semicircle and a nearly pure resistive tail at 500°C.

R. GERHARDT

1504

lE2

lE5

lE5

lE4

lE3

lE7

Frequency (Hz) Fig. 19. Tan 6, Z n and M” vs frequency plots for TiO, sample measured at 25°C. Notice the absence of a relaxation peak in any of them and also how close together they are.

due to adsorbed water and long range conduction in porous silica as well as insulating dielectric behavior and electronic conductivity in undoped titania. The realization that every dielectric function has its own relaxation time, 7 [ll], and hence can give rise to a peak in the imaginary part of the function at a different frequency for the same relaxation process, was crucial in the interpretation of the data presented. The relaxation ratio r =6Jcrn was found to have a profound effect on how far apart the different dielectric functions will appear on the frequency plane and can help determine whether the response

being measured is due to localized relaxation or long range conductivity. It should be added that caution needs to be exercised in making these assignments and that grain boundary and electrode effects (i.e. non-bulk behavior) need to be established first. This can easily be done by checking the capacitance of the process being measured, e.g. if different from the geometric capacitance then data is not representative of the bulk. Further work needs to be done to incorporate non-bulk processes with these new findings and to evaluate their effects quantitatively.

O.&-

lE2

lE3

lE4

lE5

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lE7

’

Fig. 20. Tan 6, Z” and MU vs frequency plots for TiO, sample measured at 500°C. The presence of overlapping Z Vand M’ peaks signify long range conductivity. The fastly rising tan 6 is indicative of the onset of d.c. conductivity. The extra peak in Z” at lower frequencies results from the resistive tail seen in the complex impedance plot.

Impedance and dielectric spectroscopy revisited

It was shown experimentally that in the absence of complete semicircles or when cs cannot easily be determined, frequency explicit plots are the only way to establish if more than one mechanism is operative and/or whether the data being measured is due to dipole relaxation and/or to a long range conducting or diffusing process. In particular, it was demonstrated that comparisons of the imaginary portions of the electric modulus with the dissipation factor and the imaginary impedance are extremely useful for the assignment of physical processes. Acknowledgements-The author would like to acknowledge the technical assistance from J. R. Kokan, K, Duchow, D. Wang and G. Amatucci. Former associates W. Cao and G. Zhang are particularly thanked for their contributions. The partial support from NASA/FAR under grant NAG3-1559 is gratefully acknowledged.

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