J. PRAKASH and A. K. NISHAD : ITC Parameters at the Required Heating Rate 629

phys. stat. sol. (a) 101, 629 (1987)

Subject classification: 77.30 and 77.40

Department of Physics, University of Gorakhpur')

Prediction of ITC Parameters at the Required Heating Rate BY J. PRAKASH and A. K. NISHAD

Reorientational properties of impurity-vacancy dipoles are conveniently studied by ionic thermo- current (ITC) measurements following a linear heating rate ( b ) . The ITC spectrum recorded is characterized by the ITC parameters b, T,, and I,. An effective comparison between different systems can only be established when their ITC spectra are recorded at the same b. A method is developed to predict T, and 1, a t the required value of b. The prediction suggested is found to be much useful in comparing the ITC parameters of different systems.

Umorientierungseigenschaften von Storstellen-Leerstellen-Dipolen werden durch konventionelle Messung der Ionenthermostrome (ITC), die einer linearen Heizrate ( b ) folgon, untersucht. Das aufgezeichnete ITC-Spektrum ist durch die ITC-Parameter b, T, und I , charakterisiert. Ein effektiver Vergleich zwischen unterschiedlichen Systemen l&Bt sich nur durchfiihren, wenn ihre ITC-Spektren bei demselben b aufgenommen werden. Eine Methode wird entwickelt zur Vorher- sage von T, und Im bei dem geforderten Wert von b. Die vorgeschlagene Methode erweist sich als sehr nutzlich beim Vergleich der ITC-Parameter verschiedener Systeme.

1. Introduction

It has been observed [l to 31 that with the increase in the heating rate, the peak of the ITC spectrum increases and its position simultaneously shifts towards higher tempera- ture. The ITC spectrum is very similar to a thermoluminescence glow curve with monomolecular kinetics which can be expressed as [4]

T

Q I ( T ) = -- exp d T) TP

where I (T ) is the depolarization current, E, the activation energy for the orientation of impurity-vacancy (IV) dipoles, b the linear heating rate, T p the temperature a t which the depolarization current starts to appear [5 ] , z( T ) the relaxation time a t T, z, the pre-exponential factor, k the Boltzmann constant, and Q the total charge released in ITC measurements which depends on the polarization conditions as

CXN,,U'E,A & = kTEff '

where N , is the number of IV dipoles per unit volume each with dipole moment p, E, the polarizing electric field, T2f the effective polarization temperature [6], A the effective area of the specimen, and cc a geometrical parameter which for a freely rotating dipole has the value 1/3.

The dielectric relaxation parameters E, and zo can be evaluated from the ITC spec- truni following eitherthe method suggested by Bucci et al. [a] or by Prakash et a]. [7]. The ITC spectrum happens to be an asymmetric band with its maximum depolariza-

Gorakhpur 273009, India.

630 J. PRAKASH and A. K. NISHAD

tion current (I,) a t the temperature T , such that

where z, is the relaxation time a t T , given by

Thus, the ITC spectrum can be characterized by the experimental ITC parameters b, T,Tl, and I,. It should be noted that E, and z,, are independent of b but T , and I , are affected [l, 31 by a change in b . Obviously, ITC parameters of different systems can only be compared when Tm and I, are recorded a t the same heating rate. No such systematic measurements are reported in the literature allowing such a comparison. I n this paper a method has been developed for the prediction of ITC parameters at the required value of b. It is not necessary, therefore, to record the ITC spectrum in different systems a t the same heating rate. The suggested prediction has been found to be very useful in comparing the ITC parameters of different systems.

2. ITC Parameters at the Required Heating Rate

The heating rate dependence of the peak and the position of ITC maxima is controlled through the relation [8, 91

and Q E , z0 /.̂ TI,

l n I n l = = l n - - 1--,

where I,, represents the magnitude of the maximum depolarization current appearing a t T,. These equations have been used by Prakash et al. [9] and Prakash [3] to eva- luate the dielectric relaxation parameters. It would be worth mentioning that the position of T,n is independent of the polarization conditions whereas I , depends on them through Q and ( 2 ) . Obviously, different runs for (6) must be recorded in similar polarization conditions.

A system involving a single relaxation niechanisni due to isolated IV dipoles will show a single peak in the ITC spectrum. Dielectric relaxation parameters evaluated r4, 71 from the ITC spectrum niay be helpful in knowing the value of T , a t some other heating rate with the help of ( 5 ) as [ 2 ]

where suffix 1 corresponds to the cxperimental ITC parameters and suffix 2 to the required one. Evaluation of TmZ for a required value of b, involves many steps of trial and error for the best fit of (7) . Further, the prediction of T,, for some other value of b3 needs the repetition of these clumsy and cumbersome steps. However, such a pre- diction can easily be done after rearranging (5) as

X - - l n - + l n b , E&” k I11 ~

Prediction of ITC Parameters at the Required Heating Rate 631

where Ell

k T m X , = In Tk - - . (9)

Since E, and z, are known, X , can be calculated for arbitrary values of b using (8). A graph of X , against In b will be a straight line which facilitates reading the value of X , corresponding to the required value of b . The knowledge of X , gives the value of T, using the graph of X, versus the arbitrary values of T , following (9). Thus, T, corresponding to any value of b can be determined easily.

It has been shown by van Turnhout [lo] and by Prakash et al. [9] that I, will be represented by

Using ( lo) , I, a t some other heating rate can be evaluated with the help of (6) as

I,, = I , ~ exp (2 - z)] . k Tm, Tm,

Since Tm2 is known through (8) and (9), I,, corresponding to bz can be determined. It is needless to mention that I,, will correspond to the same polarization conditions in which I,,,, has been recorded. The validity of the suggested prediction can be checked through a system where ITC spectra at different heating rates are recorded.

3. Results and Discussion

From the ITC spectrum recorded a t a particular value of 6 , ITC parameters viz. T, and I , can be predicted at any other heating rate following the suggested method. I n an attempt to know the value of T , a t the required heating rate, X, is calculated at first for arbitrary values of b using reported values of E, and z, in combination with

Fig. 1 Fig. 2

Fig. 1. Representative plot of X , against arbitrary values of b in the KI: Se2- system

Fig. 2. Representative plot of X , against arbitrary values of T, in the KI:Se2- system

63 2 J. PRAKASR and A. K. NISHAD

(8). A straight line with unit slope is obtained when X , is plotted against In b . A re- presentative straight line plot for the KI: Se2- system [11] is shown in Fig. 1. This straight line plot helps in knowing the value of Xm corresponding to the wanted value of b. Knowing X,, one gets the corresponding value of T, from a graph between X, versus T,. A representative plot of X, against the arbitrary values of Tm follow- ing (9) for KI:Se2- system [ll] is shown in Fig. 2. Thus, with the help of Fig. 1 and 2, T,, corresponding to any value of b can be predicted. Knowing T, for the wanted value of b, one can calculate the corresponding value of Im with the help of (11). Such pre- dicted values of ITC parameters a t b = 0.1 K s-l for few representative systems are presented in Table 1. It is obvious from the table that a direct comparison either with respect to the dopants or with respect to the host lattices can effectively be done when ITC parameters in different systems are known at the same heating rate. It should be remarked that a comparison of I, has no sense simply because of the fact that dif- ferent, systems have been polarized under different polarization conditions.

To have an idea about the change in T, due to a change in the activation energy of different systems, dielectric and ITC parameters of different similar dopants in the KCl

T a b l e 1 Experimental and predicted ITC parameters in different systems

system experimental ITC parameters predicted ITC parameters .~ ~ _.___ - ____.--______--

NaCI:Pb'+

KCI: Cazf

: MgS+

: SeS-

KRr:Ca2+ : Sez-

: sez- K I : Ca2

CsBr:Cazt : Pb2+

AgCl: Pb2+ :&In'+

CaF,:Gd3+ : Dy3+

SrFS:Gd3C :Dy3+

SrCI, : Gd3+ : Dy3+

BaF,: Gd3+ : K+

CdF, : Ag : Lit

0.06 0.067

0.1 0.078

0.067 0.05

0.069 0.046

0.083 0.083

0.033 0.033 0.05 0.05

0.05 0.05

0.05 0.05 0.1 0.067

0.033 0.033

228.8 206

210.5 240

205 230

197.5 215

238.5 256.5

112.3 111.0

128 126

151.1 148.3

277 270

173 175

93 85.3

2.1 x 10-14 1 .O*)

4.6*) 3.5 x 10-14

2 . 6 7 (A/cm2)

8.3 x 10-14

1.0*)

4.7*)

1.8 x 10-13 1.3 x 10-13 1.0*) 1.0*)

1.0*) 1.0")

1.0") 1.0*)

5.9 x 10-13 7.4*) 1.2 x 10-13 2.6 x 10-13

7.8 x 10-l4

6.0*)

[17] 0.1 [18] 0.1

[19] 0.1 [20] 0.1

[18] 0.1

[18] 0.1

[21] 0.1

[ll] 0.1

[22] 0.1 [22] 0.1

[l] 0.1 [l] 0.1

1231 0.1 [23] 0.1

[24] 0.1 [24] 0.1

[251 0.1 [25] 0.1

[26] 0.1 [27] 0.1

[28] 0.1 [28] 0.1

231.8 208.8

210.5 241.7

207.3 234.3

199.7 219.4

240.1 258.2

115.8 114.7 130.3 128.2

154.3 151.1

282.9 275.3

173 176.9 95.9 88.1

3.4 x 10-14 1.4*)

4.6') 4.6 x 10-14 (A/cm2) 3.8*) 16.1 x 1.4*)

5.6*) 7.2*)

15.6 x 10-14

5.1 x 10-13 3.7 x 10-13 2.0") l.Y*)

2.1") 1.9*) 1.9") 1.9") 5.9 x 10-13

3.5 x 10-13 8.0 x 10-13

10.8*)

*) R,epresents the depolarization current measured in arbitrary units.

Prediction of ITC Parameters a t the Required Heating Rate 633

Tab le 2 ITC and dielectric relaxation parameters of different divalent dopants in KCI

sample system number

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

KCl : Be2f :Niat : Mg2+ :Mnzf : 02- : Ca2+ : Ca2+ : Sr2+ :Pb2+ : Sr2+ : Yb2+ : Eu2+ : Sm2+ : Ba2+ : 52- :Sea-

-

ITC parameters a t b = 0.1 K/s

T m I m Ea TO ref.

dielectric relaxation parameters

--

(A) (arb. units) (eV) (9)

.. (K) ..__ _ _ _ ~ _

137.2 5.7 0.24 1 x ~ 1 9 1 186 3.6 0.46 1.16 X [19] 189 3.6 0.49 3.85 X [19] 192 4.3 0.49 6.2 X [19] 192.8 4.2 0.58 3.2 X [20] 210.5 4.6 0.61 3.03 X [19] 200 1 .o 0.63 7.3 X [a] 222.5 4.7 0.63 3.70 X [19] 223.5 3.7 0.65 2.08 X [19] 223 4.0 0.657 1.15 X [4] 215 3.8 0.67 1.10 X [19] 224 4.2 0.68 3.7 X [19] 227 2.0 0.70 2.70 X [19] 233 4.6 0.71 3.2 X [19] 243.9 3.1 0.77 1.3 X [20] 241.7 4.6 0.80 2.5 X [20]

lattice are presented in Table 2. To evaluate the ITC parameters of Table 2 a t b = = 0.1 K s-l, the present method of prediction has been utilized. It is apparent from the table that T, shifts towards lower temperature when E, decreases, provided the heating rate is maintained a t the same value. The dependence of T, on E, can be as- sessed through Fig. 3 where In T, is plotted against In E, using the data of divalent dopants in KCl a t b = 0.1 K s-1 presented in Table 2. The slope of the straight line comes out to be 0.5.Such a dependence of T, on Ea can be expressed after rearranging (3) as

(12) 1 bz, 1 In T, = - In - f- In E, . 2 E 2

Fig. 3. Plot of In T, against I n Ea in KCI lattice for the data of different divalent dopants recorded a t 6 = 0.1 Ks-l

634 J. PRAKASH and A. K. NISHAD

Fig. 3 in combination with (12) suggests that z, should be fixed for a particular host lattice when the data of different similar dopants recorded a t the same value of b are considered, A similar behaviour has also been observed in other host lattices. Thus, the suggested method of prediction is very useful in comparing the ITC parameters of different systems.

The claim about the correctness of the predicted parameters following the suggested method will be ascertained through the ITC data of a system recorded at different heating rates. Such data are fortunately available in the literature on a couple of sys- tems namely AgCl : Ni2+ [ 11 ; KC1 : ASOF, KBr : AsO:, and K I : AsOF [3] ; (3-metal free phthalocyanine (P-H,Pc) [12] ; ice [13] ; amorphous semiconductors [14] ; poylvinyl butyral films [15], and SiO, films [16]. The predicted values of Tm along with the ex- perimental ones recorded a t different heating rates for these systems are presented in Table 3. A very good agreement has been observed between the experimental and predicted values of T,. It obviously testifies the validity of the suggested method for predicting the ITC parameters a t the required heating rate. A slight difference between the experimental and the predicted values of T, observed at higher heating rates is primarily due to the onset of temperature gradient and thermal lag across the speci- men. I n case of p-H,Pc the reported value [12] of the activation energy is 0.37 eV. With this value of E,, the predicted values of T, are found to be off from the corre- sponding experimental values as obvious from Table 3. A straight line plot of In (T&/b) against 1/T, for B-H,Pc as per (5) is shown in Fig. 4. The slope of the line gives a value of 0.29 eV for E,. Using this value of E,, the predicted values of T, are found to be in very good agreement with those recorded experimentally. It seems, therefore, quite reasonable that 0.29 eV is the correct value of the activation energy for P-H2Pc in- stead of the reported value of 0.37 eV. Similar is the case with ice (see Table 3). I n

Table 3 Predicted and experimental values of T, at different heating rates

~ ~~ ~~ ~

system E, b Tm (K) ref. (eV) (K s-l) experimental predicted

0.162

0.169

0.183

AgCl : Ni2f 0.339 0.033 0.05 0.067 0.10 0.15

KCl : AsO; 36.00 x 101.32 X

KBr : AsO, 8.33 x 10-3 14.35 x 10-3 19.17 x 10-8 29.86 x 50.00 x 69.20 x

KI: AsO; 7.07 x 10-3 8.57 x 10-3 9.55 x 10-3

16.11 x 21.25 x 27.78 x 38.33 x

113.1 114.3 115.5 117.4 120.4

66.50 58.26 58.75 59.25 60.00 60.50 61.50 62.50 59.25 59.50 60.00 60.50 61.00 61.50 62.00

113.3 P I 114.6 P I 115.5 c11 116.8 c11 118.2 111

56.50 [31 58.30 [31 58.35 c31 59.25 [31 59.75 [31 60.50 131 61.45 c31 62.05 [31 59.25 c31 59.55 ~ 3 1 59.70 c31 60.55 [31 61.05 [31 61.45 [31 62.00 c31

Prediction of ITC Parameters a t the Required Heating Rate 635

Tab 1 e 3 (continued)

P-HZPc 0.37 (reported)

0.29

ice 0.17 In (T&/b) vs. 1/Tm

0.25 (initial rise method)

0.189

amorphous semiconductor (i) MSIM 0.59

structure

(ii) SMIM 0.60 structure

PVB film

2.02

SiO, 0.19 film

0.05 0.15 0.25 0.50 0.05 0.15 0.25 0.50 0.012 0.020 0.042 0.058 0.087 0.192 0.012 0.020 0.042 0.058 0.087 0.192 0.012 0.020 0.042 0.058 0.087 0.192

0.01 0.06 0.11 0.16 0.21

0.01 0.06 0.11 0.16 0.21 0.016 0.032 0.065 0.082 0.099 0.132

0.24 0.39 0.56 0.78 1.04

exaerimental aredicted

252.2 271.8 283.3 295.6 252.2 271.8 283.3 295.6 120.3 123.3 127.9 129.9 135.3 139.1 120.3 123.3 127.9 129.9 135.3 139.1 120.3 123.3 127.9 129.9 135.3 139.1

182.4 191.2 194.4 196.3 198.5

337.2 365.7 376.5 383.6 388.4 325.6 326.7 329.0 330.5 331.6 333.3 233.6 244.0 252.5 261.0 267.3

256 271 280 292 252 272 282 296 120.0 123.7 128.0 131.5 134.8 141.5 120.5 122.5 126.5 128.0 130.0 134.5 120.3 123.3 128.0 130.2 133.0 139.0

182.4 191.2 194.2 196.4 197.8

337.0 365.5 376.5 383.5 389.0 323.5 326.7 329.8 331.0 331.8 333.0 233.8 244.0 252.5 260.5 267.5 -~

12 physica (a) 101/2

63 6 J. PRAKASR and A. K. NISHAD

the latter case, the reported values[l3] of E , are 0.25eV from the initial rise method and 0.17 eV from the In (T:/b) versus l / T , plot. With these values of E,, the predicted and the experimental values of T, (see Table 3) differ. Based on the data presented in Table 3, a plot of In (TL/b) against l/Tm for ice results in a straight line similar to Fig. 4. The slope of the line gives a value of 0.189 eV for E,. Using this value of E,, the predicted and the experimental values of T, are found to be in reasonably good agreement. Thus, 0.189 eV seems to be the correct value of E, for ice. The extent of agreement between the experimental and the predicted parameters lends, therefore, a testimony to the utility of the present method in the cases where ITC and dielectric relaxation parameters are to be compared.

Acknowledgements

The authors are thankful to Prof. N. K. Sanyal, Prof. F. Pischer (Miinster, FRG), and Prof. J. Vanderschueren (IAge, Belgium) for valuable suggestions. One of the author (AKN) is thankful to C.S.I.R. (New Delhi) for financial assistance.

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Prediction of ITC Parameters at the Required Heating Rate 637

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Materials, Vol. 1, Ed. A. BISHAY, Plenum Publ. Co., New York 1974 (p. 15).

(Received February 24,1987)

42 *