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Analysis of space charge relaxation by equivalent electric circuit for yttrium doped CaF2
PCS 1552
Journal of Physics and Chemistry of Solids 60 (1999) 247–256
Analysis of space charge relaxation by equivalent electric circuit for yttrium doped CaF2 Kazuaki Tanaka, Katsumi Tanaka, Katsuyasu Kawano, Ryouhei Nakata* Department of Electronic Engineering, Denki-Tsuushin University, Chofu-shi, Tokyo 182-8585, Japan Received 16 May 1997; accepted 8 July 1998
Abstract Characteristic behaviours of high temperature ionic thermocurrent (HT ITC) in CaF2:Y were studied by a usual ITC method with Teflon insulated electrodes in the nominal yttrium concentration range from n 0.002 to 30 mol%. An equivalent electric circuit was used to analyze HT ITC by assuming a space charge flow model with a specific resistivity in the form of r (T) r syso T exp(Ea/kT). The analysis revealed that a relaxation of the space charges occurred in the specific resistivity from insulating to semi-insulating change-over of CaF2:Y crystal. In addition, dependences of the peak temperature shift and total charge of HT ITC on the crystal thickness could be explained by taking into account the capacitance of the crystal. And the yttrium concentration dependence of the activation energy for the space charge flow could be well approximated by Ea(n) Eao ⫺ g n 1/3. 䉷 1998 Elsevier Science Ltd. All rights reserved. Keywords: A. Inorganic compounds; D. Defects; D. Electrical properties
1. Introduction After the first application of an ionic thermocurrent (ITC) method to studies of electric dipole disorientation in alkali halides by Bucci and Fieschi [1], this method has been used to study high temperature ionic thermocurrent (HT ITC) in alkaline earth fluorides doped with rare earth (RE) impurities. Many researchers have attempted to prove the origin of the HT ITC band [2–10] [11–20] [21–28]. In a previous paper, we reported two characteristics of the HT ITC band in BaF2:Y studied by a thermal sampling method or a temperature window method using Teflon insulated electrodes [29]. One characteristic is that no remarkable polarizable ITC bands could be observed in the temperature range above the HT ITC band, though many kinds of polarizable defects or clusters exist in the crystal. The other characteristic is that a relation between the temperature of the HT ITC peak and the polarization temperature could separate the intense HT ITC band from additional (polarizable) HT bands appearing at the low temperature tail of the intense one, i.e. the observed HT ITC band consists of at least two kinds of * Corresponding author
HT ITC sources. The intense HT ITC spectrum due to the space charge (mobile interstitial fluoride ions) could be characterized by a simple relaxation time t (T) through
t T to T exp Ea =kT
1
where t o (s/K) is the constant, Ea the activation energy of a mobile interstitial fluoride ion Fi⫺ , and k the Boltzmann constant. Although relations between temperature of HT ITC peak (Tm) and rare earth concentration (n) were reported on many alkaline earth fluorides especially for SrF2 [6–10] [11,12,14–17,20,22,23], Ea versus rare earth concentration relations were not studied in detail except for those estimated with the ionic conductivity and ITC methods for SrF2:Yb [9], the ITC method for SrF2:La [10], the AC dielectric loss method for SrF2:Ce [23] and the ITC method for BaF2:Y [29]. It was reported that Ea in BaF2:Y varied linearly with one-third powers of nominal yttrium concentration (n 1/3) below the several mol%. And this linear relation could be apparently explained by the Lidiard–Debye– Hu¨ckel (LDH) model, suggesting that the doped yttrium impurities in BaF2 form statistical defects or clusters. We
0022-3697/99/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(98)00268-6
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K. Tanaka et al. / Journal of Physics and Chemistry of Solids 60 (1999) 247–256 Table 1 Derived parameters for the intense HT ITC bands of CaF2:Y. ‘s’ in t syso (s/K) stands for second n (mol%)
Tm (K)
Ea (eV)
t syso (s/K)
0.002 0.005 0.01 0.05 0.1 0.5 1 2 3 4 5 7 10 13 20 30
393 388 385 386 382 393 391 393 385 375 361 338 315 292 270 241
1.237 1.235 1.220 1.198 1.218 1.198 1.041 0.940 0.980 0.881 0.922 0.750 0.693 0.634 0.556 0.450
5.11 × 10 ⫺17 3.25 × 10 ⫺17 3.91 × 10 ⫺17 8.84 × 10 ⫺17 3.21 × 10 ⫺17 1.67 × 10 ⫺16 1.70 × 10 ⫺14 4.25 × 10 ⫺13 7.00 × 10 ⫺14 6.98 × 10 ⫺13 6.08 × 10 ⫺14 3.38 × 10 ⫺12 4.50 × 10 ⫺12 6.09 × 10 ⫺12 2.37 × 10 ⫺11 2.53 × 10 ⫺10
2. Experimental procedure
Fig. 1. Experimental HT ITC spectra observed in CaF2 doped with different yttrium concentration. (a) HT ITC spectra for n 0.002 to 0.5 nominal mol% and (b) for 1.0 to 30 mol%. On details of Tm, see Table 1. (c) Simulated and experimental spectra for 1.0, 10 and 20 mol% shown with broken and solid lines, respectively.
are interested in whether or not such a linear relation between Ea and n 1/3 holds for CaF2:Y. In this paper we report on experimental characteristics of HT ITC such as Ea versus n 1/3 in CaF2:Y. Next we analyze the influence of the crystal thickness, the Teflon foils and our ITC measurement system on the characteristics of Tm and total charges related to HT ITC by using a simple equivalent electric circuit.
Single crystals used in our ITC experiment were meltgrown in vacuum by the Bridgman Stockbarger method. The starting materials were reagent grade powders of CaF2 and YF3. And small amount of PbF2 powder was added to prevent the crystals from hydration and oxidation [30]. Details of the crystal growth were published elsewhere [27]. As-grown crystals were cylindrical in shape with about 8 mm in diameter and 30–35 mm in length. For HT ITC experiments, sample crystals were cut from the crystal boules with wire saw and finished to a disk shape with 2 mm in thickness by fine polishing powder. We prepared 16 samples with different nominal concentration of n 0.002, 0.005, 0.01, 0.05, 0.10, 0.50, 1.00, 2.00, 3.00, 4.00, 5.00, 7.00, 10.0, 13.0, 20.0 and 30.0 mol%. The density measurement by a micro-balance showed that the densities of CaF2 for n ⬎ 3.0 mol% agreed, to a good approximation, with the reported ones [31]. For n ⱕ 3 mol%, we use the nominal concentration in a following HT ITC analysis for the present. A set of ITC measurements was made with a vibrating reed electrometer (Advantest TR84M). The temperature of the crystal was controlled between 100 and 450 K by a personal computer. The polarization voltage and time were 2000 V and 3 min, respectively. The polarization temperature (Tp) was usually chosen between 410 and 430 K, considering the temperature Tm of the HT ITC peak. The heating rate b dT/dt was 0.079 K/s. As stated above, Teflon insulated electrodes were used in order to decrease spontaneous current due to chemical reaction on the surfaces and/or charge injections from electrodes into the sample crystal.
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Fig. 2. Peak temperature (Tm) versus n 1/3. Experimental results for CaF2:Y are shown with open circles, while those for CaF2:Eu are shown with asterisks as a reference.
Fig. 4. Total HT ITC charges versus n 1/3 for CaF2:Y. Charges deduced from experimental and simulated HT ITC spectra are shown with large and small open circles, respectively.
3. Experimental results and analysis
in Fig. 1 and is depicted in Fig. 4 with large empty circles. Like that of the maxima of HT ITC in Fig. 3, the total charge tends to decrease with increasing yttrium concentration. The full width of half maximum (FWHM) of the HT ITC band is shown with large circles in Fig. 5. The FWHM increases up to n 1/3 1 (n 1.0 mol%) and then decreases, showing the growth of polarizable defects at the rising tail of the intense HT ITC band below n 1/3 1.0. In our previous HT ITC analyses [28,29], HT ITC parameters of t o and Ea were determined by taking no account of the influence due to Teflon foils, lead wires and the electrometer for our ITC measurement system. Here we derive the intrinsic HT ITC parameters of CaF2:Y crystals by including the measurement system. In a following HT ITC analysis we use a HT ITC formulae for the space charge flow in the crystal which is given by the following form:
Experimental ITC spectra are shown in Fig. 1 with solid lines. A weak ITC band around 130 K (Fig. 1(a)) was already ascribed to the reorientation or disorientation of the nearest neighbour (NN) Y 3⫹ ⫺ Fi⫺ [2]. Furthermore, there appear two unidentified ITC bands or shoulders around 310 and 360 K (Fig. 1(a) and (b)). On the other hand, the intense ITC band, which Tm depends on yttrium concentration, appears in the high temperature spectral region of Fig. 1. And no remarkable ITC bands can be seen above Tm of the intense HT ITC band. The temperatures of peak HT ITC observed in different yttrium concentration are listed in Table 1. The Tm –n 1/3 relation is plotted in Fig. 2 with open circles where we used n 1/3 in place of n [29]. It is clear that Tm increases slightly with yttrium concentration up to n 1/3 1.3 (n 2 mol%) and then decreases monotonically. The maxima of HT ITC are shown with circles in Fig. 3 and have a tendency to decrease with increasing yttrium concentration. The total charge (QHT) related to the HI ITC band was estimated by integrating each HT ITC band
Fig. 3. Maximum HT ITC (ITCm) versus n 1/3.
I T QHT =tsys exp⫺b⫺1
Z
dT 0 =tsys
2
and a relaxation time (t sys(T)) of the total system is assumed
Fig. 5. Full width of half maximum (FWHM) versus n 1/3. FWHMs derived from experimental and simulated HT ITC spectra are shown with large and small circles, respectively.
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Fig. 8. Equivalent electric circuit used to analyze the characteristics of HT ITC. On Ri, Ci, CT1, Ccry, Rcry, CT3 and Q in the figure, see text. Fig. 6. Activation energies (Ea) versus n 1/3. Open circles are for CaF2:Y and asterisks are for CaF2:Eu. A straight line is the leastmean-square fit for CaF2:Y.
by
tsys T tsyso T exp Ea =kT
3
where QHT is the total charge, t syso the preexponential factor, and T just before exp(Ea/kT) comes from the temperature dependence of the resistivity r (T) due to the migration of interstitial fluoride ions in the crystal. By differentiating eqn (2) with respect to T and using eqn (3), the preexponential factor t syso is related to Ea through
tsyso kTm exp ⫺Ea =kTm =b Ea ⫺ kTm
4
Each of the experimental spectra was fitted to simulated ones of eqn (2) by substituting the experimental Tm in Table 1 into eqn (4) and varying Ea from 0.4 and 1.5 eV with 0.001 eV steps. The best fit values of t syso and Ea are listed in Table 1. Some of the simulated spectra are shown in Fig. 1(c) with broken lines for n 1.0, 10 and 20 nominal mol%. It is clear that the experimental spectra are in good agreement with the simulated ones over the central and high temperature regions of the HT ITC bands, but they deviate from the simulated ones at the low temperature tails,
Fig. 7. Relaxation time (t sys(Tm)) versus n 1/3 for the total measurement system.
probably due to polarizable defects. Empty circles in Fig. 6 show the Ea –n 1/3 relation. For low yttrium concentration n 1/3 ⬍ 1, Ea shown with filled circles is nearly constant like that of CaF2:Eu, suggesting an isolated preferential clustering formation [20,29]. And the least-mean-square fit of Ea is expressed by a solid line: Ea Eao ⫺ 0:276 × n1=3 eV
5
where Eao is the activation energy of the space charge flow (the interstitial fluoride ion flow) across the crystal at n 1/3 0 and is estimated to be 1.31 eV. The relaxation time of the overall system at Tm is given by
tsys Tm tsyso Tm exp Ea =kTm
6
and the experimental t sys(Tm) versus n 1/3 is shown in Fig. 7. It should be noticed that the derived t syso and t sys(Tm) are related with the capacitance of the crystal, the Teflon foils, the electrometer and the signal lead wires together with the resistance of the crystal and the input resistance of the electrometer. Hereafter, we state how to estimate the intrinsic HT ITC parameters of the crystal by using an equivalent electric circuit. It is reasonable, from a physical point of view, to approximate an equivalent electric circuit for the space charge relaxation by a parallel connection of a capacitance (Ccry) and a temperature dependent resistance (Rcry). This circuit is the simplest equivalent circuit given by Mu¨ller for semi-insulating materials [32]. For a simplicity of calculation for the HT ITC parameters of the crystal, the input resistance (Ri) of the electrometer should be taken as low as possible and was put to be 10 10 V here. The maximum input capacitance cei of the electrometer is 15 pF, according to the technical manual. In addition, maximum stray capacitance was estimated to be about 15 pF by considering an insulating Teflon foil inserted between one of the signal electrode and the spring-loaded metal supporter, the length of the signal lead wires, etc. Then the overall maximum input capacitance (Ci) was finally estimated to be about 30 pF. The other important values are the capacitance of the Teflon foil 1 (CT1) and foil 2 (CT2) inserted between the crystal and the electrodes. The capacitance of Teflon foil condensers (CT1 CT2) is also
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polarization voltage (Vp 2000 V) is impressed across the two Teflon foils, but not across the crystal [16,32]. Thus each of the Teflon foil condensers is charged by an amount Qo Vp = 1=CT1 ⫹ 1=CT2
9
This charge is held constant when the temperature of the crystal is lowered under application of the polarization voltage if the crystal and Teflon foil have no temperature dependent dielectric constants. After removing the polarization voltage at low temperature, the lead wires are shortcircuited for 30 min to remove the charges left on the lead wires, etc. Then the redistributed charge on the Teflon foil condensers (CT1 and CT2) becomes Fig. 9. Capacitance (Ccry) versus n 1/3 for CaF2:Y. Large and small circles are estimated by integrating experimental and simulated HT ITC spectra, respectively.
estimated by neglecting the fringing effect. Using the thickness (0.1 mm), the diameter (8 mm) of the Teflon foil and the static dielectric constant (e 2) [32], CT1 becomes about 8.9 (pF). Thus the equivalent electric circuit for the total system is finally given in Fig. 8. The relaxation time (t cry) of the crystal is expressed by the product of Ccry and Rcry and is
tcry Ccry Rcry
7
and Rcry Rcryo T exp Eca =kT
8
where Ccry is assumed to be independent of temperature, Rcryo is the preexponential factor and has a dimension of [V/K]. It is obvious that the activation energy Eca is equal to Ea in eqn (3). Thus t sys(T) is expressed in a form similar to eqn (1). We derive Ccry and Rcry from t cry in eqn (8). At polarization temperature Tp (Tp ⬎ Tm), the resistance of the Teflon foil is much higher than that of Rcry. Then almost all the
QT1 0 QT2 0 Qo =Ccry 1=CT1 ⫹ 1=Ccry ⫹ 1=CT2 10 and the charge on the crystal (Ccry) is Qc 0 ⫺Qo 1=CT1 ⫹ 1=CT2 = 1=CT1 ⫹ 1=Ccry ⫹ 1=CT2 11 When eqn (10) is solved with respect to the Ccry, we obtain Ccry Qo =QT1 0 ⫺ 1= 1=CT1 ⫹ 1=CT2 Qo =QT1 0 ⫺ 1Qo =Vp
12
Then approximate Ccry can be estimated, when QT1(0) is replaced by the experimental QHT. And the derived Ccry values are shown in Fig. 9 with large empty circles. We can see from this figure that the Ccry increases with yttrium concentration. From Fig. 7, the relaxation time at Tm is about 160 s or approximately f 1/160 Hz. Considering the rounded shape of each HT ITC spectrum, the impedance (Z) of a parallel connection of Ri and Ci can be approximated by Z 1= 1=Ri ⫹ j2pfCi 1= 10⫺10 ⫹ j1:2 × 10⫺12 ⱌ 1010 V
13
Thus the input impedance is treated as pure resistance. The resistance Rcry of the crystal in Fig. 8 is calculated readily when the input impedance of the electrometer is much lower than Rcry around Tm. From this equivalent circuit, t sys is expressed by
tsys Rcry Ccry ⫹ CT1 CT2 = CT1 ⫹ CT2
14
Then Rcry(Tm) is obtained by Rcry T tsys T= Ccry ⫹ CT1 CT2 = CT1 ⫹ CT2
Fig. 10. Specific resistance r (Tm) of the crystal versus n 1/3. Large and small open circles are estimated from the experimental and stimulated HT ITC, respectively.
15
Substituting t sys(Tm) in Table 1 into eqn (15), Rcry(Tm) becomes about 2 × 10 13 V which clearly satisfies the inequality of Rcry(Tm) q Ri. The calculated resistivity r (Tm) is shown in Fig. 10 with large empty circles and ranges from 4 to 8 × 10 13 V cm. In the above calculation we used QHT in place of QT1 as the first approximation, because we were not aware of the dependence of Ccry on yttrium concentration. We found in this approximate calculation that Ci and Ri could be
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After a little manipulation, eqns (16), (17), (18), (19) are reduced to dQi =dt ⫺Qi = Ci Ri ⫺ dQT1 =dt
20
dQc =dt ⫺Qc = Ccry Rcry ⫹ dQT1 =dt
21
QT1 CT1 Qi =Ci ⫺ Qcry =Cc =2
22
where CT1 CT2 and QT1 QT2.These equations are solved numerically under the initial conditions given by eqns (10), (11) and a following condition at t 0, i.e. Ci has no charges at t 0: Fig. 11. Experimental and simulated HT ITC spectra for (a) n 0.002, (b) 3.0, and (c) 30 mol%. Solid and broken lines are experimental and simulated spectra, respectively.
neglected in our ITC measurement system when Ri 10 10 V. We will show that the use of the equivalent circuit is a useful and simple means to analyze characteristic behaviours of HT ITC bands related with a space charge flow. In a following calculation, both Ci and Ri are taken into account, though they are negligible as stated just above. The inflow of HT ITC into the electrometer is expressed by ITC Qi = Ci Ri ⫺dQi =dt ⫺ dQT1 =dt
16
The current through the Teflon foil is the sum of the current of the crystal, Ic and Ir, ⫺dQT1 =dt ⫺dQc =dt ⫹ Ir
17
Equating the voltage across Ccry to that across the resistor Rcry, the current Ir through the Rcry becomes Ir ⫺Qc = Ccry Rcry
18
Finally the voltage across Ci is equal to the sum of those of other series-connected capacitors, we get Qi =Ci QT1 =CT1 ⫹ Qc =Ccry ⫹ QT2 =CT2
19
Qi 0 0
23
The HT ITC spectra were calculated by using eqn (16) and were fitted to those of the experimental ones in Fig. 1 from which Ccry –n 1/3 and Rcryo –n 1/3 relations could be derived. The Ccry –n 1/3 relation is shown in Fig. 9 with small empty circles, indicating that Ccry increases also with increasing yttrium concentration. The specific resistivity of r (Tm) of the crystal at Tm is derived from the following equation and is shown in Fig. 10 with small empty circles:
r Tm Rcry Tm S=d
24
where S and d are the cross-sectional area and the thickness of the crystal sample, respectively. The simulated spectra for n 0.002, 3.00 and 30 mol% yttrium concentration are shown in Fig. 11 with broken lines. These spectra are also in good agreement with the experimental ones for the central and high temperature regions of the HT ITC bands. There is, however, some disagreement in the simulated and experimental spectra at the low temperature tails of the intense HT ITC ones, suggesting the existence of some additional (polarizable) relaxation bands. In addition, the t cry(Tm)–n 1/3 relation is shown in Fig. 12. It is clear that t cry(Tm) is smaller than t sys(Tm) as shown in Fig. 7. This fact indicates that the Teflon foil condensers or capacitance between the electrodes and the crystal give a substantial influence on the characteristic parameters of the HT ITC spectra. 4. Discussion
Fig. 12. Relaxation time (t cry(Tm)) versus n 1/3 for CaF2:Y crystals.
In line with previous experimental procedures performed in BaF2:Y [29], we derived the relaxation parameters (t syso and Ea) of the intense HT ITC band in CaF2:Y by fitting the simulated HT ITC spectra to the experimental ones over the central and high temperature spectral regions. The intrinsic parameter of t cryo was derived from t syso with the aid of the equivalent electric circuit in Fig. 8. Subsequently Ccry in eqn (12) and Rcry in eqn (15) were determined. In the above calculation, we used the nominal yttrium concentration for n ⱕ 3 mol%. The ITC band due to the orientation of the nearest neighbour (NN) dipoles was clearly observed as
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Rcryo 2 1:05 × 10⫺2 V=K
26
tcryo 2 2:34 × 10⫺14 s=K
27
and Eca Ea 0:98 eV
28
In eqns (25), (26), (27), ‘2’ in parentheses stands for the sample thickness of ‘2’ mm. The simulation was performed under the same experimental conditions stated previously and two assumptions as follows. The capacitance of the sample with d (mm) in thickness is given by Fig. 13. Dependence of HT ITC spectra on crystal thickness (d) with Vp 2000 V and the HT ITC parameters or n 3.0 mol% yttrium concentration in Table 1. From left to right peaks, the crystal thickness is d 0.3, 0.5, 1.0, 2.0, 4.0 and 8.0 mm, respectively.
shown with an arrow in Fig. 1(a). And the magnitude of ITC increased up to n 0.1 mol% with increasing yttrium concentration and then decreased. The concentration of n 0.1 mol% giving rise to maximum NN ITC agrees well with previously reported results for CaF2:RE [26,34]. Furthermore, it was reported that yttrium impurities were dissolved into CaF2 crystals with nearly nominal yttrium concentration [33]. From these facts, the use of nominal concentration seems to give no substantial errors in our calculated results except for very low yttrium concentration. Next it is needed to justify the use of the equivalent circuit in Fig. 8, by which the following characteristics of HT ITC in alkaline earth fluorides doped with constant rare earth (RE) concentration should be explained: (1) the magnitude of HT ITC or QHT are proportional to applied electric field (Vp); (2) the maximum magnitude of HT ITC increases with increasing sample thickness (d); and (3) Tm shifts to high temperature with increasing sample thickness as observed in SrF2:Gd [16]. For the first characteristic, it is obvious from eqns (9), (10) and the results for BaF2:Y [29]. We simulate HT ITC spectra related with the second and third characteristics by using the derived parameters for the crystal doped with 3.0 mol% yttrium concentration and with 2 mm in thickness Ccry 2 2:23 pF
25
Ccry d Ccry 2 2=d
29
and the resistance Rcry(d) is Rcry d Rcry 2 d=2
30
In the simulation, we use eqns (20), (21), (22). The simulated spectra are shown in Fig. 13. These results are semiquantitatively consistent with the second and third characteristics of HT ITC. In addition, it is also shown from eqns (9), (10) that the maximum magnitude of HT ITC or QT1 saturates as Ccry decrease, i.e. the thickness of the crystal increases. It was suggested from eqn (13) that the simulation of the HT ITC spectra could be expressed in a simple equivalent circuit if Ri 10 10 V (Ri p Rcrys(Tm)) was assumed. When we use this condition, the relaxation time of the total system with the Teflon insulated electrodes can be written in a following form
tsys d ⱌ Rcry dCcry d ⫹ CT1 CT2 = CT1 ⫹ CT2 ⱌ Rcry 2Ccry 2 ⫹ Rcry 2CT1 CT2 = CT1 ⫹ CT2 d=2 31 It is clear that t sys(d) increases with d, i.e. Tm is shifted to high temperature with increasing the sample thickness. In other words, t sys(d) increases with decreasing the thickness of the Teflon foils. From these facts, it is worth noting that the parameter t syso of HT ITC is strongly influenced by ‘effective capacitance’ between the crystal and electrodes even if insulating electrodes are not used. In any way, the use of the equivalent circuit is a powerful means for HT ITC analyses of the space charge flow relaxation. The crystal thickness dependence is shown in Table 2.
Table 2 Crystal thickness dependence of t cryo, Tm, ITCm, QT1 and t cry(Tm) for n 3 mol% yttrium d (mm)
Ea (eV)
t cryo (s/K)
0.3 0.5 1.0 2.0 4.0 8.0
0.98 0.98 0.98 0.98 0.98 0.98
2.34 × 2.34 × 2.34 × 2.34 × 2.34 × 2.34 ×
10 ⫺14 10 ⫺14 10 ⫺14 10 ⫺14 10 ⫺14 10 ⫺14
Tm (K)
ITCm (pA)
QT1 (pC)
t cry (Tm) (s)
375 376 380 385 392 400
5.19 7.36 10.65 13.61 15.59 16.52
2.21 × 10 3 3.16 × 10 3 4.66 × 10 3 6.12 × 10 3 7.25 × 10 3 7.99 × 10 3
125 108 82 55 34 19
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The equivalent circuit analysis of the HT ITC spectra showed a large difference in the simulated and experimental spectra at the low temperature tails for n 0.002 and 3.0 mol% as shown in Fig. 11. A large difference in the experimental (large circles) and simulated (small circles) FWHMs for n 1/3 0.46 to 1.4 or n 0.1 to 3 mol% in Fig. 5 is also responsible for the existence of many polarizable defects contributing to the broadening of the HT ITC band. The band around 360 K in Fig. 1(a) may be due to polarizable defects formed during the crystal growth because this band cannot be seen in the experimental spectrum for n 1 mol% in Fig. 1(c). In addition, other weak ITC bands are always observed only at the low temperature tail of the intense HT ITC band by the small difference in the simulated and experimental spectra for n 1.0, 10 and 20 mol% in Fig. 1(c) and n 30 mol% in Fig. 11. This result suggests that, irrespective of doped yttrium concentration, potentially polarizable defects or clusters (latent polarizable defects or clusters) exist in the crystal. Similar weak ITC bands were observed in BaF2:Y [29] and were distinguished from that of the intense HT ITC band by the temperature window method. And the Tm values of the weak bands in the Tm –Tp relation decreased to that of the polarizable NN or NNN dipoles along one steep slope straight line with increasing yttrium concentration. Furthermore, the smooth change-over from the intense HT ITC band to the weak bands was observed in Tm. Then the weak ITC bands in the low temperature tail are candidates for a new intense HT ITC band, when yttrium concentration is increased. The origin of the weak bands was a dipole-like relaxation or hopping relaxation of a space charge [23], where latent polarizable defects provide a path of a space charge flow. Many types of nonpolarizable defects or clusters were reported [24], some of which may participate in HT ITC. The capacitance Ccry in Fig. 8 or explicitly in eqn (12) was introduced into the analysis of HT ITC through the total charge (QHT or QT1(0)) of HT ITC. And we derived Ccry by using the charges in Fig. 4. The large and small circles in Fig. 9 are the experimental capacitance and the calculated capacitance from the simulated HT ITC curves, respectively. And the differences in large and small circles in Ccry are ascribed to the charges due to the additional ITC bands at the low temperature tail of the experimental intense HT ITC band. Irrespective of two kinds of the empty circles, the derived capacitance Ccry in Fig. 9 increases with increasing yttrium concentration and the maximum Ccry is about two or three times as large as that estimated from e 6.799 [35] (shown with a filled circle in the same figure). In other words, e of CaF2:Y is increased with increasing yttrium concentration within our experimental limit. Then it is concluded that the possible candidate of the increase in e is yttrium impurities like those of erbium and lanthanum impurities in CaF2 as measured with the dielectric loss method [36]. Besides, the e contributing to the increase of Ccry is coming from defects polarized at T ⬃ 85 K. It is, however, unknown whether the increase of e is caused by
the polarized defects on the surface or inside the bulk of the CaF2:Y crystal. Similar increase of e due to polarizable defects at low temperature was found with dielectric loss measurements in SrF2:Ce [23]. The large increase of e in CaF2:Y is, of course, not expected by the electronic polarization of doped YF3. In any way, the decrease of QHT with increasing n 1/3 in Fig. 4 is strongly related to the increase of e . The magnitude of the specific resistance r in insulating materials is important in a space charge flow relaxation, because the relaxation occurs in a specific resistance ranging from insulating (r ⬃ 10 15 V cm) to semi-insulating (10 13 V cm) change-over [32]. As shown in Fig. 10, the r (Tm) of CaF2:Y are placed in this change-over range and support the space charge flow model for our intense HT ITC. Using the ionic conductivity curve in the range of s T ⬎ 10 ⫺6 (V ⫺1 cm ⫺1 K) for n 0.01 mol% by Bollmann et al. for CaF2:Y [37], the resistivity (r ) at T Tm (385 K or n 0.01 mol% in Table 1) was estimated by extrapolating their s T versus 10 000/T relation and was about 1.5 × 10 12 (V cm). Considering the different experimental methods, the resistivity of r 7 × 10 13 (V cm) in Fig. 10 is rather in good agreement with the above 1.5 × 10 12 (V cm). An experimental fact that Tm .5 of CaF2:Y like CaF2:Eu hold constant below n 1/3 1.26 or n 2 mol% in Fig. 2 suggests a preferential defect or cluster formation at low RE concentration [20,29], consistent with the behaviours of the FWHM around n 1/3 1 to 3 in Fig. 5. And the horizontal region of the Ea –n 1/3 for n 1/3 ⬍ 1 as shown with filled circles in Fig. 6 seems also to be related to the preferential clusters. However, the derived Ea for CaF2:Y fall on the straight line and are different from the Ea values for CaF2:Eu. Using the circles for CaF2:Y in Fig. 6, the least-mean-square fit gave Eao 1.31 eV (Eao Ea at n 0) which is smaller than 1.4 eV as reported in CaF2:Eu [28]. The difference in those Eao values may probably depend on the difference of the preferential cluster formation for trivalent yttrium and europium impurities in CaF2. The value of Eao 1.31 eV is rather close to 1.32 eV in the association range of CaF2:Y by Bollmann et al. [37] than 1.24 eV in the dissociation range of CaF2:Y by Jacobs et al. [38]. The activation energy of Ea 1.24 eV in the temperature range from 588 to 833 K was ascribed to dissociated Fi⫺ from clusters (dimers) by DC ionic conductivity. This temperature range is, of course, far above our Tm in Table 1. Then it is natural to conclude that our activation energy Eao 1.31 eV is one for the association range, consistent with r 7 × 10 13 (V cm). The accurate Eao should be systematically studied in connection with preferential clusters in CaF2 doped with very low RE concentration. The decrease of the activation energy (DEa) for low RE concentration has been usually explained by the Lidiard– Debye–Hu¨ckel (LDH) theory [29,39,40] and is given by DEa ⫺e2 k=e 1 ⫹ kR
32
K. Tanaka et al. / Journal of Physics and Chemistry of Solids 60 (1999) 247–256
Fig. 14. Calculated DEa versus n 1/3 by using the Lidiard–Debye– Hu¨ckel theory for R 0.65Rm. Large and small open circles are experimental and calculated values, respectively. The straight line is the least-mean-square fit of the small circles.
with
k2 8pNe2 n= 100 ekT
33 3⫹
where R is the radius of the virtual sphere centred on Y ion in which one yttrium ion is involved [29] and N 2.45 × 10 22 cm ⫺3 the site numbers of Ca 2⫹ ions in unit volume. If trivalent yttrium ions are homogeneously dispersed in CaF2, the maximum radius (Rm) of the sphere and yttrium concentration (n) in mol% are related by Rm 3=2p1=3 dF⫺F n=100⫺1=3
34
where dF–F is one half of the lattice constant of CaF2. The experimental DEa values are plotted in Fig. 14 with large empty circles where Eao 1.31 eV was used. And the calculated DEa for T 430 K could be fitted to experimental ones and are shown with small empty circles or a solid line in Fig. 14 by choosing R 0.65Rm. And the DEa is approximated, for large k R or low yttrium concentration, by DEa n 0:012 ⫺ 0:280 n1=3
35
where we neglected the influence of doped yttrium concentration on the expansion of CaF2 cell and the dielectric constant e (6.799 [35]). From the experimental results of DEa in Fig. 14, the D–n 1/3 relation is approximately linear over the wide range from n 1/3 1 to 3.1 like that observed in BaF2:Y [29]. This fact indicates that a space charge relaxes via the same relaxation mechanism in CaF2:Y. The observed wide linear range is, however, abnormal, because the LDH theory is only valid for the intermediate RE concentration range [37]. Different from the preferential cluster formation range for n 1/3 ⬍ 1 in Fig. 6, ITC due to NN Y 3⫹ ⫺ Fi⫺ dipoles observed in Fig. 1(a) is extinguished in the intermediate range in Fig. 1(b). This evolution has been usually explained by the interaction of one NN Y 3⫹ ⫺ Fi⫺ dipole with neighbouring NN dipoles [16] or clustering of NN Y 3⫹ ⫺ Fi⫺ dipoles. In fact, many types of yttrium defects
255
were observed such as 2:2:2 clusters (dimers) studied by neutron diffraction [33] and by calculation [42], gettered 2:2:2 clusters by AC dielectric loss and calculation [36], and Y 2⫹ ions in CaF2 by ESR [41]. And isolated Y 3⫹ ions like cubic Gd 3⫹ ions [44] may possibly act as the charged defect incorporated in the crystal growth. Taking into account the conversion from the latent polarizable defects to the weakly polarized ones observed as the weak ITC bands at the rising part of the intense HT ITC band, one of the possible candidates of the defects is latent polarizable dimers constructed by two NN dipoles [36,42]. And such dimers are expected to be homogeneously dispersed in the crystal to maintain the LDH theory. Combining the reported results in the thermal sampling ITC method [29] with an equivalent electric circuit analysis, we analyzed the behaviours of HT ITC in CaF2:Y. Two important results of HT ITC were found in CaF2:Y when Teflon insulated electrodes were used. One is t sys(Tm) which is two or three times larger than t cry(Tm). The other is Tm which is strongly dependent on the capacitance of the Teflon foils and the sample crystal. Accordingly, the activation energy Ea is a reliable HT ITC parameter for HT ITC.
References [1] C. Bucci, R. Fieschi, Phys. Rev. Lett. 12 (1964) 16. [2] I. Kunze, P. Mu¨ller, Phys. Stat. Sol. (a) 13 (1972) 197. [3] E. Laredo, M. Puma, D.R. Figueroa, Phys. Rev. B19 (1979) 2224. [4] E. Laredo, D.R. Figueroa, M. Puma, J. Phys. (Paris) 41 (1980) C6–451. [5] M. Puma, E. Laredo, M.E. Galavı´s, D.R. Figueroa, Phys. Rev. B22 (1980) 5791. [6] H.W. den Hartog, J.C. Langevoort, Phys. Rev. B24 (1981) 3547. [7] J. Meuldijk, H.H. Mulder, H.W. den Hartog, Phys. Rev. B25 (1982) 5204. [8] D. Figueroa, E. Laredo, M. Puma, N. Sua´rez, Crystal Lattice Defects 9 (1982) 167. [9] J. Meuldijk, H.W. den Hartog, Phys. Rev. B27 (1983) 6376. [10] J. Meuldijk, H.W. den Hartog, Phys. Rev. B28 (1983) 1036. [11] E. Laredo, M. Puma, N. Sua´rez, D. Figueroa, Solid State Ionics 9 and 10 (1983) 497. [12] J. Meuldijk, G. Kiers, H.W. den Hartog, Phys. Rev. B28 (1983) 6022. [13] H.W. den Hartog, K.F. Pen, J. Meuldijk, Phys. Rev. B28 (1983) 6031. [14] H.W. den Hartog, R. Nakata, J. Phys. Soc. Jap. 52 (1983) 3110. [15] J. Meuldijk, R. van der Meulen, H.W. den Hartog, Phys. Rev. B29 (1984) 2153. [16] P. Dorenbos, H.W. den Hartog, Phys. Rev. B31 (1985) 3932. [17] L. Puma, A. Bello, N. Sua´rez, E. Laredo, Phys. Rev. B32 (1985) 5424. [18] E. Laredo, N. Sua´rez, A. Bello, M. Puma, D. Figueroa, J. Schoonman, Phys. Rev. B32 (1985) 8325. [19] H.W. den Hartog, P. Dorebos, W.J. Postma, Phys. Rev. B34 (1986) 7496.
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[20] P. Dorenbos, G. Kruizinga, H.W. den Hartog, Cryst. Latt. Def. Amorph. Mat. 15 (1987) 217. [21] E. Laredo, N. Sua´rez, M.E. Galavis, M. Puma, A. Bello, D. Figueroa, Crys. Latt. Def. Amorph. Mat. 15 (1987) 271. [22] P. Dorenbos, S. Vrind, J. Dolfing, H.W. den Hartog, Phys. Rev. B35 (1987) 5766. [23] P. Dorenbos, A. van der Lee, H.W. den Hartog, Phys. Rev. B37 (1988) 10312. [24] E. Laredo, N. Sua´rez, A. Bello, M. Puma, Solid State Ionics 37 (1990) 103. [25] N. Sua´rez, E. Laredo, F. Lorenzo, A. Bello, M. Puma, Solid State Ionics 38 (1990) 63. [26] E. Laredo, M. Diaz, N. Sua´rez, A. Bello, Phys. Rev. B46 (1992) 11415. [27] Y. Matsumoto, K. Tanaka, K. Kawano, R. Nakata, M. Sumita, J. Phys. Chem. Solids 54 (1993) 331. [28] K. Tanaka, Q. Zhuang, K. Tanaka, K. Kawano, R. Nakata, J. Phys. Chem. Solids 57 (1996) 307. [29] F. Uchiyama, E. Ishikawa, K. Tanaka, K. Kawano, R. Nakata, J. Phys. Chem. Solids 58 (1997) 2007. [30] B.P.M. Lenting, J.A.J. Numan, E.J. Bijvank, H.W. den Hartog, Phys. Rev. B14 (1976) 1811. [31] J. Short, R. Roy, J. Phys. Chem. 67 (1963) 1860. [32] P. Mu¨ller, Phys. Stat. Sol. (a) 67 (1981) 11.
[33] A.K. Cheetham, B.E.F. Fender, M.J. Cooper, J. Phys. C: Solid State Phys. 4 (1971) 3107. [34] R. Capelletti, E. Okuno, G.E. Matthews, J.H. Crawford Jr., Phys. Stat. Sol. (a) 47 (1978) 617. [35] C. Andeen, J. Fontanella, D. Schuele, Rev. Sci. Inst. 41 (1970) 1573. [36] C.G. Andeen, J.J. Fontanella, M.C. Wintersgill, P.J. Welcher, R.J. Kimble Jr., G.E. Matthews Jr., J. Phys. C: Solid State Phys. 14 (1981) 3557. [37] W. Bollmann, P. Go¨rlich, W. Hauk, H. Mothes, Phys. Stat. Sol. (a) 2 (1970) 157. [38] P.W.M. Jacobs, S.H. Ong, J. Phys. Chem. Solids 41 (1980) 431. [39] R.G. Fuller, in: J.H. Crawford Jr., L.M. Slifkin (Eds.), Point Defects in Solids, Vol. 1, Chap. 2, Plenum Press, New York and London, 1972. [40] K.E.D. Wapenaar, J.L. van Koesveld, J. Schoonman, Solid State Ionics 2 (1981) 145. [41] J.R. O’Connor, J.H. Chen, Appl. Phys. Lett. 5 (1964) 100. [42] J. Corish, C.R.A. Catlow, P.W.M. Jacobs, S.H. Ong, Phys. Rev. B25 (1982) 6425. [43] E.L. Kitts Jr., J.H. Crawford Jr., Phys. Rev. B9 (1974) 5264. [44] W. Low, Phys. Rev. 109 (1958) 265.
Journal of Physics and Chemistry of Solids 60 (1999) 247–256
Analysis of space charge relaxation by equivalent electric circuit for yttrium doped CaF2 Kazuaki Tanaka, Katsumi Tanaka, Katsuyasu Kawano, Ryouhei Nakata* Department of Electronic Engineering, Denki-Tsuushin University, Chofu-shi, Tokyo 182-8585, Japan Received 16 May 1997; accepted 8 July 1998
Abstract Characteristic behaviours of high temperature ionic thermocurrent (HT ITC) in CaF2:Y were studied by a usual ITC method with Teflon insulated electrodes in the nominal yttrium concentration range from n 0.002 to 30 mol%. An equivalent electric circuit was used to analyze HT ITC by assuming a space charge flow model with a specific resistivity in the form of r (T) r syso T exp(Ea/kT). The analysis revealed that a relaxation of the space charges occurred in the specific resistivity from insulating to semi-insulating change-over of CaF2:Y crystal. In addition, dependences of the peak temperature shift and total charge of HT ITC on the crystal thickness could be explained by taking into account the capacitance of the crystal. And the yttrium concentration dependence of the activation energy for the space charge flow could be well approximated by Ea(n) Eao ⫺ g n 1/3. 䉷 1998 Elsevier Science Ltd. All rights reserved. Keywords: A. Inorganic compounds; D. Defects; D. Electrical properties
1. Introduction After the first application of an ionic thermocurrent (ITC) method to studies of electric dipole disorientation in alkali halides by Bucci and Fieschi [1], this method has been used to study high temperature ionic thermocurrent (HT ITC) in alkaline earth fluorides doped with rare earth (RE) impurities. Many researchers have attempted to prove the origin of the HT ITC band [2–10] [11–20] [21–28]. In a previous paper, we reported two characteristics of the HT ITC band in BaF2:Y studied by a thermal sampling method or a temperature window method using Teflon insulated electrodes [29]. One characteristic is that no remarkable polarizable ITC bands could be observed in the temperature range above the HT ITC band, though many kinds of polarizable defects or clusters exist in the crystal. The other characteristic is that a relation between the temperature of the HT ITC peak and the polarization temperature could separate the intense HT ITC band from additional (polarizable) HT bands appearing at the low temperature tail of the intense one, i.e. the observed HT ITC band consists of at least two kinds of * Corresponding author
HT ITC sources. The intense HT ITC spectrum due to the space charge (mobile interstitial fluoride ions) could be characterized by a simple relaxation time t (T) through
t T to T exp Ea =kT
1
where t o (s/K) is the constant, Ea the activation energy of a mobile interstitial fluoride ion Fi⫺ , and k the Boltzmann constant. Although relations between temperature of HT ITC peak (Tm) and rare earth concentration (n) were reported on many alkaline earth fluorides especially for SrF2 [6–10] [11,12,14–17,20,22,23], Ea versus rare earth concentration relations were not studied in detail except for those estimated with the ionic conductivity and ITC methods for SrF2:Yb [9], the ITC method for SrF2:La [10], the AC dielectric loss method for SrF2:Ce [23] and the ITC method for BaF2:Y [29]. It was reported that Ea in BaF2:Y varied linearly with one-third powers of nominal yttrium concentration (n 1/3) below the several mol%. And this linear relation could be apparently explained by the Lidiard–Debye– Hu¨ckel (LDH) model, suggesting that the doped yttrium impurities in BaF2 form statistical defects or clusters. We
0022-3697/99/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(98)00268-6
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K. Tanaka et al. / Journal of Physics and Chemistry of Solids 60 (1999) 247–256 Table 1 Derived parameters for the intense HT ITC bands of CaF2:Y. ‘s’ in t syso (s/K) stands for second n (mol%)
Tm (K)
Ea (eV)
t syso (s/K)
0.002 0.005 0.01 0.05 0.1 0.5 1 2 3 4 5 7 10 13 20 30
393 388 385 386 382 393 391 393 385 375 361 338 315 292 270 241
1.237 1.235 1.220 1.198 1.218 1.198 1.041 0.940 0.980 0.881 0.922 0.750 0.693 0.634 0.556 0.450
5.11 × 10 ⫺17 3.25 × 10 ⫺17 3.91 × 10 ⫺17 8.84 × 10 ⫺17 3.21 × 10 ⫺17 1.67 × 10 ⫺16 1.70 × 10 ⫺14 4.25 × 10 ⫺13 7.00 × 10 ⫺14 6.98 × 10 ⫺13 6.08 × 10 ⫺14 3.38 × 10 ⫺12 4.50 × 10 ⫺12 6.09 × 10 ⫺12 2.37 × 10 ⫺11 2.53 × 10 ⫺10
2. Experimental procedure
Fig. 1. Experimental HT ITC spectra observed in CaF2 doped with different yttrium concentration. (a) HT ITC spectra for n 0.002 to 0.5 nominal mol% and (b) for 1.0 to 30 mol%. On details of Tm, see Table 1. (c) Simulated and experimental spectra for 1.0, 10 and 20 mol% shown with broken and solid lines, respectively.
are interested in whether or not such a linear relation between Ea and n 1/3 holds for CaF2:Y. In this paper we report on experimental characteristics of HT ITC such as Ea versus n 1/3 in CaF2:Y. Next we analyze the influence of the crystal thickness, the Teflon foils and our ITC measurement system on the characteristics of Tm and total charges related to HT ITC by using a simple equivalent electric circuit.
Single crystals used in our ITC experiment were meltgrown in vacuum by the Bridgman Stockbarger method. The starting materials were reagent grade powders of CaF2 and YF3. And small amount of PbF2 powder was added to prevent the crystals from hydration and oxidation [30]. Details of the crystal growth were published elsewhere [27]. As-grown crystals were cylindrical in shape with about 8 mm in diameter and 30–35 mm in length. For HT ITC experiments, sample crystals were cut from the crystal boules with wire saw and finished to a disk shape with 2 mm in thickness by fine polishing powder. We prepared 16 samples with different nominal concentration of n 0.002, 0.005, 0.01, 0.05, 0.10, 0.50, 1.00, 2.00, 3.00, 4.00, 5.00, 7.00, 10.0, 13.0, 20.0 and 30.0 mol%. The density measurement by a micro-balance showed that the densities of CaF2 for n ⬎ 3.0 mol% agreed, to a good approximation, with the reported ones [31]. For n ⱕ 3 mol%, we use the nominal concentration in a following HT ITC analysis for the present. A set of ITC measurements was made with a vibrating reed electrometer (Advantest TR84M). The temperature of the crystal was controlled between 100 and 450 K by a personal computer. The polarization voltage and time were 2000 V and 3 min, respectively. The polarization temperature (Tp) was usually chosen between 410 and 430 K, considering the temperature Tm of the HT ITC peak. The heating rate b dT/dt was 0.079 K/s. As stated above, Teflon insulated electrodes were used in order to decrease spontaneous current due to chemical reaction on the surfaces and/or charge injections from electrodes into the sample crystal.
K. Tanaka et al. / Journal of Physics and Chemistry of Solids 60 (1999) 247–256
249
Fig. 2. Peak temperature (Tm) versus n 1/3. Experimental results for CaF2:Y are shown with open circles, while those for CaF2:Eu are shown with asterisks as a reference.
Fig. 4. Total HT ITC charges versus n 1/3 for CaF2:Y. Charges deduced from experimental and simulated HT ITC spectra are shown with large and small open circles, respectively.
3. Experimental results and analysis
in Fig. 1 and is depicted in Fig. 4 with large empty circles. Like that of the maxima of HT ITC in Fig. 3, the total charge tends to decrease with increasing yttrium concentration. The full width of half maximum (FWHM) of the HT ITC band is shown with large circles in Fig. 5. The FWHM increases up to n 1/3 1 (n 1.0 mol%) and then decreases, showing the growth of polarizable defects at the rising tail of the intense HT ITC band below n 1/3 1.0. In our previous HT ITC analyses [28,29], HT ITC parameters of t o and Ea were determined by taking no account of the influence due to Teflon foils, lead wires and the electrometer for our ITC measurement system. Here we derive the intrinsic HT ITC parameters of CaF2:Y crystals by including the measurement system. In a following HT ITC analysis we use a HT ITC formulae for the space charge flow in the crystal which is given by the following form:
Experimental ITC spectra are shown in Fig. 1 with solid lines. A weak ITC band around 130 K (Fig. 1(a)) was already ascribed to the reorientation or disorientation of the nearest neighbour (NN) Y 3⫹ ⫺ Fi⫺ [2]. Furthermore, there appear two unidentified ITC bands or shoulders around 310 and 360 K (Fig. 1(a) and (b)). On the other hand, the intense ITC band, which Tm depends on yttrium concentration, appears in the high temperature spectral region of Fig. 1. And no remarkable ITC bands can be seen above Tm of the intense HT ITC band. The temperatures of peak HT ITC observed in different yttrium concentration are listed in Table 1. The Tm –n 1/3 relation is plotted in Fig. 2 with open circles where we used n 1/3 in place of n [29]. It is clear that Tm increases slightly with yttrium concentration up to n 1/3 1.3 (n 2 mol%) and then decreases monotonically. The maxima of HT ITC are shown with circles in Fig. 3 and have a tendency to decrease with increasing yttrium concentration. The total charge (QHT) related to the HI ITC band was estimated by integrating each HT ITC band
Fig. 3. Maximum HT ITC (ITCm) versus n 1/3.
I T QHT =tsys exp⫺b⫺1
Z
dT 0 =tsys
2
and a relaxation time (t sys(T)) of the total system is assumed
Fig. 5. Full width of half maximum (FWHM) versus n 1/3. FWHMs derived from experimental and simulated HT ITC spectra are shown with large and small circles, respectively.
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Fig. 8. Equivalent electric circuit used to analyze the characteristics of HT ITC. On Ri, Ci, CT1, Ccry, Rcry, CT3 and Q in the figure, see text. Fig. 6. Activation energies (Ea) versus n 1/3. Open circles are for CaF2:Y and asterisks are for CaF2:Eu. A straight line is the leastmean-square fit for CaF2:Y.
by
tsys T tsyso T exp Ea =kT
3
where QHT is the total charge, t syso the preexponential factor, and T just before exp(Ea/kT) comes from the temperature dependence of the resistivity r (T) due to the migration of interstitial fluoride ions in the crystal. By differentiating eqn (2) with respect to T and using eqn (3), the preexponential factor t syso is related to Ea through
tsyso kTm exp ⫺Ea =kTm =b Ea ⫺ kTm
4
Each of the experimental spectra was fitted to simulated ones of eqn (2) by substituting the experimental Tm in Table 1 into eqn (4) and varying Ea from 0.4 and 1.5 eV with 0.001 eV steps. The best fit values of t syso and Ea are listed in Table 1. Some of the simulated spectra are shown in Fig. 1(c) with broken lines for n 1.0, 10 and 20 nominal mol%. It is clear that the experimental spectra are in good agreement with the simulated ones over the central and high temperature regions of the HT ITC bands, but they deviate from the simulated ones at the low temperature tails,
Fig. 7. Relaxation time (t sys(Tm)) versus n 1/3 for the total measurement system.
probably due to polarizable defects. Empty circles in Fig. 6 show the Ea –n 1/3 relation. For low yttrium concentration n 1/3 ⬍ 1, Ea shown with filled circles is nearly constant like that of CaF2:Eu, suggesting an isolated preferential clustering formation [20,29]. And the least-mean-square fit of Ea is expressed by a solid line: Ea Eao ⫺ 0:276 × n1=3 eV
5
where Eao is the activation energy of the space charge flow (the interstitial fluoride ion flow) across the crystal at n 1/3 0 and is estimated to be 1.31 eV. The relaxation time of the overall system at Tm is given by
tsys Tm tsyso Tm exp Ea =kTm
6
and the experimental t sys(Tm) versus n 1/3 is shown in Fig. 7. It should be noticed that the derived t syso and t sys(Tm) are related with the capacitance of the crystal, the Teflon foils, the electrometer and the signal lead wires together with the resistance of the crystal and the input resistance of the electrometer. Hereafter, we state how to estimate the intrinsic HT ITC parameters of the crystal by using an equivalent electric circuit. It is reasonable, from a physical point of view, to approximate an equivalent electric circuit for the space charge relaxation by a parallel connection of a capacitance (Ccry) and a temperature dependent resistance (Rcry). This circuit is the simplest equivalent circuit given by Mu¨ller for semi-insulating materials [32]. For a simplicity of calculation for the HT ITC parameters of the crystal, the input resistance (Ri) of the electrometer should be taken as low as possible and was put to be 10 10 V here. The maximum input capacitance cei of the electrometer is 15 pF, according to the technical manual. In addition, maximum stray capacitance was estimated to be about 15 pF by considering an insulating Teflon foil inserted between one of the signal electrode and the spring-loaded metal supporter, the length of the signal lead wires, etc. Then the overall maximum input capacitance (Ci) was finally estimated to be about 30 pF. The other important values are the capacitance of the Teflon foil 1 (CT1) and foil 2 (CT2) inserted between the crystal and the electrodes. The capacitance of Teflon foil condensers (CT1 CT2) is also
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251
polarization voltage (Vp 2000 V) is impressed across the two Teflon foils, but not across the crystal [16,32]. Thus each of the Teflon foil condensers is charged by an amount Qo Vp = 1=CT1 ⫹ 1=CT2
9
This charge is held constant when the temperature of the crystal is lowered under application of the polarization voltage if the crystal and Teflon foil have no temperature dependent dielectric constants. After removing the polarization voltage at low temperature, the lead wires are shortcircuited for 30 min to remove the charges left on the lead wires, etc. Then the redistributed charge on the Teflon foil condensers (CT1 and CT2) becomes Fig. 9. Capacitance (Ccry) versus n 1/3 for CaF2:Y. Large and small circles are estimated by integrating experimental and simulated HT ITC spectra, respectively.
estimated by neglecting the fringing effect. Using the thickness (0.1 mm), the diameter (8 mm) of the Teflon foil and the static dielectric constant (e 2) [32], CT1 becomes about 8.9 (pF). Thus the equivalent electric circuit for the total system is finally given in Fig. 8. The relaxation time (t cry) of the crystal is expressed by the product of Ccry and Rcry and is
tcry Ccry Rcry
7
and Rcry Rcryo T exp Eca =kT
8
where Ccry is assumed to be independent of temperature, Rcryo is the preexponential factor and has a dimension of [V/K]. It is obvious that the activation energy Eca is equal to Ea in eqn (3). Thus t sys(T) is expressed in a form similar to eqn (1). We derive Ccry and Rcry from t cry in eqn (8). At polarization temperature Tp (Tp ⬎ Tm), the resistance of the Teflon foil is much higher than that of Rcry. Then almost all the
QT1 0 QT2 0 Qo =Ccry 1=CT1 ⫹ 1=Ccry ⫹ 1=CT2 10 and the charge on the crystal (Ccry) is Qc 0 ⫺Qo 1=CT1 ⫹ 1=CT2 = 1=CT1 ⫹ 1=Ccry ⫹ 1=CT2 11 When eqn (10) is solved with respect to the Ccry, we obtain Ccry Qo =QT1 0 ⫺ 1= 1=CT1 ⫹ 1=CT2 Qo =QT1 0 ⫺ 1Qo =Vp
12
Then approximate Ccry can be estimated, when QT1(0) is replaced by the experimental QHT. And the derived Ccry values are shown in Fig. 9 with large empty circles. We can see from this figure that the Ccry increases with yttrium concentration. From Fig. 7, the relaxation time at Tm is about 160 s or approximately f 1/160 Hz. Considering the rounded shape of each HT ITC spectrum, the impedance (Z) of a parallel connection of Ri and Ci can be approximated by Z 1= 1=Ri ⫹ j2pfCi 1= 10⫺10 ⫹ j1:2 × 10⫺12 ⱌ 1010 V
13
Thus the input impedance is treated as pure resistance. The resistance Rcry of the crystal in Fig. 8 is calculated readily when the input impedance of the electrometer is much lower than Rcry around Tm. From this equivalent circuit, t sys is expressed by
tsys Rcry Ccry ⫹ CT1 CT2 = CT1 ⫹ CT2
14
Then Rcry(Tm) is obtained by Rcry T tsys T= Ccry ⫹ CT1 CT2 = CT1 ⫹ CT2
Fig. 10. Specific resistance r (Tm) of the crystal versus n 1/3. Large and small open circles are estimated from the experimental and stimulated HT ITC, respectively.
15
Substituting t sys(Tm) in Table 1 into eqn (15), Rcry(Tm) becomes about 2 × 10 13 V which clearly satisfies the inequality of Rcry(Tm) q Ri. The calculated resistivity r (Tm) is shown in Fig. 10 with large empty circles and ranges from 4 to 8 × 10 13 V cm. In the above calculation we used QHT in place of QT1 as the first approximation, because we were not aware of the dependence of Ccry on yttrium concentration. We found in this approximate calculation that Ci and Ri could be
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After a little manipulation, eqns (16), (17), (18), (19) are reduced to dQi =dt ⫺Qi = Ci Ri ⫺ dQT1 =dt
20
dQc =dt ⫺Qc = Ccry Rcry ⫹ dQT1 =dt
21
QT1 CT1 Qi =Ci ⫺ Qcry =Cc =2
22
where CT1 CT2 and QT1 QT2.These equations are solved numerically under the initial conditions given by eqns (10), (11) and a following condition at t 0, i.e. Ci has no charges at t 0: Fig. 11. Experimental and simulated HT ITC spectra for (a) n 0.002, (b) 3.0, and (c) 30 mol%. Solid and broken lines are experimental and simulated spectra, respectively.
neglected in our ITC measurement system when Ri 10 10 V. We will show that the use of the equivalent circuit is a useful and simple means to analyze characteristic behaviours of HT ITC bands related with a space charge flow. In a following calculation, both Ci and Ri are taken into account, though they are negligible as stated just above. The inflow of HT ITC into the electrometer is expressed by ITC Qi = Ci Ri ⫺dQi =dt ⫺ dQT1 =dt
16
The current through the Teflon foil is the sum of the current of the crystal, Ic and Ir, ⫺dQT1 =dt ⫺dQc =dt ⫹ Ir
17
Equating the voltage across Ccry to that across the resistor Rcry, the current Ir through the Rcry becomes Ir ⫺Qc = Ccry Rcry
18
Finally the voltage across Ci is equal to the sum of those of other series-connected capacitors, we get Qi =Ci QT1 =CT1 ⫹ Qc =Ccry ⫹ QT2 =CT2
19
Qi 0 0
23
The HT ITC spectra were calculated by using eqn (16) and were fitted to those of the experimental ones in Fig. 1 from which Ccry –n 1/3 and Rcryo –n 1/3 relations could be derived. The Ccry –n 1/3 relation is shown in Fig. 9 with small empty circles, indicating that Ccry increases also with increasing yttrium concentration. The specific resistivity of r (Tm) of the crystal at Tm is derived from the following equation and is shown in Fig. 10 with small empty circles:
r Tm Rcry Tm S=d
24
where S and d are the cross-sectional area and the thickness of the crystal sample, respectively. The simulated spectra for n 0.002, 3.00 and 30 mol% yttrium concentration are shown in Fig. 11 with broken lines. These spectra are also in good agreement with the experimental ones for the central and high temperature regions of the HT ITC bands. There is, however, some disagreement in the simulated and experimental spectra at the low temperature tails of the intense HT ITC ones, suggesting the existence of some additional (polarizable) relaxation bands. In addition, the t cry(Tm)–n 1/3 relation is shown in Fig. 12. It is clear that t cry(Tm) is smaller than t sys(Tm) as shown in Fig. 7. This fact indicates that the Teflon foil condensers or capacitance between the electrodes and the crystal give a substantial influence on the characteristic parameters of the HT ITC spectra. 4. Discussion
Fig. 12. Relaxation time (t cry(Tm)) versus n 1/3 for CaF2:Y crystals.
In line with previous experimental procedures performed in BaF2:Y [29], we derived the relaxation parameters (t syso and Ea) of the intense HT ITC band in CaF2:Y by fitting the simulated HT ITC spectra to the experimental ones over the central and high temperature spectral regions. The intrinsic parameter of t cryo was derived from t syso with the aid of the equivalent electric circuit in Fig. 8. Subsequently Ccry in eqn (12) and Rcry in eqn (15) were determined. In the above calculation, we used the nominal yttrium concentration for n ⱕ 3 mol%. The ITC band due to the orientation of the nearest neighbour (NN) dipoles was clearly observed as
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253
Rcryo 2 1:05 × 10⫺2 V=K
26
tcryo 2 2:34 × 10⫺14 s=K
27
and Eca Ea 0:98 eV
28
In eqns (25), (26), (27), ‘2’ in parentheses stands for the sample thickness of ‘2’ mm. The simulation was performed under the same experimental conditions stated previously and two assumptions as follows. The capacitance of the sample with d (mm) in thickness is given by Fig. 13. Dependence of HT ITC spectra on crystal thickness (d) with Vp 2000 V and the HT ITC parameters or n 3.0 mol% yttrium concentration in Table 1. From left to right peaks, the crystal thickness is d 0.3, 0.5, 1.0, 2.0, 4.0 and 8.0 mm, respectively.
shown with an arrow in Fig. 1(a). And the magnitude of ITC increased up to n 0.1 mol% with increasing yttrium concentration and then decreased. The concentration of n 0.1 mol% giving rise to maximum NN ITC agrees well with previously reported results for CaF2:RE [26,34]. Furthermore, it was reported that yttrium impurities were dissolved into CaF2 crystals with nearly nominal yttrium concentration [33]. From these facts, the use of nominal concentration seems to give no substantial errors in our calculated results except for very low yttrium concentration. Next it is needed to justify the use of the equivalent circuit in Fig. 8, by which the following characteristics of HT ITC in alkaline earth fluorides doped with constant rare earth (RE) concentration should be explained: (1) the magnitude of HT ITC or QHT are proportional to applied electric field (Vp); (2) the maximum magnitude of HT ITC increases with increasing sample thickness (d); and (3) Tm shifts to high temperature with increasing sample thickness as observed in SrF2:Gd [16]. For the first characteristic, it is obvious from eqns (9), (10) and the results for BaF2:Y [29]. We simulate HT ITC spectra related with the second and third characteristics by using the derived parameters for the crystal doped with 3.0 mol% yttrium concentration and with 2 mm in thickness Ccry 2 2:23 pF
25
Ccry d Ccry 2 2=d
29
and the resistance Rcry(d) is Rcry d Rcry 2 d=2
30
In the simulation, we use eqns (20), (21), (22). The simulated spectra are shown in Fig. 13. These results are semiquantitatively consistent with the second and third characteristics of HT ITC. In addition, it is also shown from eqns (9), (10) that the maximum magnitude of HT ITC or QT1 saturates as Ccry decrease, i.e. the thickness of the crystal increases. It was suggested from eqn (13) that the simulation of the HT ITC spectra could be expressed in a simple equivalent circuit if Ri 10 10 V (Ri p Rcrys(Tm)) was assumed. When we use this condition, the relaxation time of the total system with the Teflon insulated electrodes can be written in a following form
tsys d ⱌ Rcry dCcry d ⫹ CT1 CT2 = CT1 ⫹ CT2 ⱌ Rcry 2Ccry 2 ⫹ Rcry 2CT1 CT2 = CT1 ⫹ CT2 d=2 31 It is clear that t sys(d) increases with d, i.e. Tm is shifted to high temperature with increasing the sample thickness. In other words, t sys(d) increases with decreasing the thickness of the Teflon foils. From these facts, it is worth noting that the parameter t syso of HT ITC is strongly influenced by ‘effective capacitance’ between the crystal and electrodes even if insulating electrodes are not used. In any way, the use of the equivalent circuit is a powerful means for HT ITC analyses of the space charge flow relaxation. The crystal thickness dependence is shown in Table 2.
Table 2 Crystal thickness dependence of t cryo, Tm, ITCm, QT1 and t cry(Tm) for n 3 mol% yttrium d (mm)
Ea (eV)
t cryo (s/K)
0.3 0.5 1.0 2.0 4.0 8.0
0.98 0.98 0.98 0.98 0.98 0.98
2.34 × 2.34 × 2.34 × 2.34 × 2.34 × 2.34 ×
10 ⫺14 10 ⫺14 10 ⫺14 10 ⫺14 10 ⫺14 10 ⫺14
Tm (K)
ITCm (pA)
QT1 (pC)
t cry (Tm) (s)
375 376 380 385 392 400
5.19 7.36 10.65 13.61 15.59 16.52
2.21 × 10 3 3.16 × 10 3 4.66 × 10 3 6.12 × 10 3 7.25 × 10 3 7.99 × 10 3
125 108 82 55 34 19
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The equivalent circuit analysis of the HT ITC spectra showed a large difference in the simulated and experimental spectra at the low temperature tails for n 0.002 and 3.0 mol% as shown in Fig. 11. A large difference in the experimental (large circles) and simulated (small circles) FWHMs for n 1/3 0.46 to 1.4 or n 0.1 to 3 mol% in Fig. 5 is also responsible for the existence of many polarizable defects contributing to the broadening of the HT ITC band. The band around 360 K in Fig. 1(a) may be due to polarizable defects formed during the crystal growth because this band cannot be seen in the experimental spectrum for n 1 mol% in Fig. 1(c). In addition, other weak ITC bands are always observed only at the low temperature tail of the intense HT ITC band by the small difference in the simulated and experimental spectra for n 1.0, 10 and 20 mol% in Fig. 1(c) and n 30 mol% in Fig. 11. This result suggests that, irrespective of doped yttrium concentration, potentially polarizable defects or clusters (latent polarizable defects or clusters) exist in the crystal. Similar weak ITC bands were observed in BaF2:Y [29] and were distinguished from that of the intense HT ITC band by the temperature window method. And the Tm values of the weak bands in the Tm –Tp relation decreased to that of the polarizable NN or NNN dipoles along one steep slope straight line with increasing yttrium concentration. Furthermore, the smooth change-over from the intense HT ITC band to the weak bands was observed in Tm. Then the weak ITC bands in the low temperature tail are candidates for a new intense HT ITC band, when yttrium concentration is increased. The origin of the weak bands was a dipole-like relaxation or hopping relaxation of a space charge [23], where latent polarizable defects provide a path of a space charge flow. Many types of nonpolarizable defects or clusters were reported [24], some of which may participate in HT ITC. The capacitance Ccry in Fig. 8 or explicitly in eqn (12) was introduced into the analysis of HT ITC through the total charge (QHT or QT1(0)) of HT ITC. And we derived Ccry by using the charges in Fig. 4. The large and small circles in Fig. 9 are the experimental capacitance and the calculated capacitance from the simulated HT ITC curves, respectively. And the differences in large and small circles in Ccry are ascribed to the charges due to the additional ITC bands at the low temperature tail of the experimental intense HT ITC band. Irrespective of two kinds of the empty circles, the derived capacitance Ccry in Fig. 9 increases with increasing yttrium concentration and the maximum Ccry is about two or three times as large as that estimated from e 6.799 [35] (shown with a filled circle in the same figure). In other words, e of CaF2:Y is increased with increasing yttrium concentration within our experimental limit. Then it is concluded that the possible candidate of the increase in e is yttrium impurities like those of erbium and lanthanum impurities in CaF2 as measured with the dielectric loss method [36]. Besides, the e contributing to the increase of Ccry is coming from defects polarized at T ⬃ 85 K. It is, however, unknown whether the increase of e is caused by
the polarized defects on the surface or inside the bulk of the CaF2:Y crystal. Similar increase of e due to polarizable defects at low temperature was found with dielectric loss measurements in SrF2:Ce [23]. The large increase of e in CaF2:Y is, of course, not expected by the electronic polarization of doped YF3. In any way, the decrease of QHT with increasing n 1/3 in Fig. 4 is strongly related to the increase of e . The magnitude of the specific resistance r in insulating materials is important in a space charge flow relaxation, because the relaxation occurs in a specific resistance ranging from insulating (r ⬃ 10 15 V cm) to semi-insulating (10 13 V cm) change-over [32]. As shown in Fig. 10, the r (Tm) of CaF2:Y are placed in this change-over range and support the space charge flow model for our intense HT ITC. Using the ionic conductivity curve in the range of s T ⬎ 10 ⫺6 (V ⫺1 cm ⫺1 K) for n 0.01 mol% by Bollmann et al. for CaF2:Y [37], the resistivity (r ) at T Tm (385 K or n 0.01 mol% in Table 1) was estimated by extrapolating their s T versus 10 000/T relation and was about 1.5 × 10 12 (V cm). Considering the different experimental methods, the resistivity of r 7 × 10 13 (V cm) in Fig. 10 is rather in good agreement with the above 1.5 × 10 12 (V cm). An experimental fact that Tm .5 of CaF2:Y like CaF2:Eu hold constant below n 1/3 1.26 or n 2 mol% in Fig. 2 suggests a preferential defect or cluster formation at low RE concentration [20,29], consistent with the behaviours of the FWHM around n 1/3 1 to 3 in Fig. 5. And the horizontal region of the Ea –n 1/3 for n 1/3 ⬍ 1 as shown with filled circles in Fig. 6 seems also to be related to the preferential clusters. However, the derived Ea for CaF2:Y fall on the straight line and are different from the Ea values for CaF2:Eu. Using the circles for CaF2:Y in Fig. 6, the least-mean-square fit gave Eao 1.31 eV (Eao Ea at n 0) which is smaller than 1.4 eV as reported in CaF2:Eu [28]. The difference in those Eao values may probably depend on the difference of the preferential cluster formation for trivalent yttrium and europium impurities in CaF2. The value of Eao 1.31 eV is rather close to 1.32 eV in the association range of CaF2:Y by Bollmann et al. [37] than 1.24 eV in the dissociation range of CaF2:Y by Jacobs et al. [38]. The activation energy of Ea 1.24 eV in the temperature range from 588 to 833 K was ascribed to dissociated Fi⫺ from clusters (dimers) by DC ionic conductivity. This temperature range is, of course, far above our Tm in Table 1. Then it is natural to conclude that our activation energy Eao 1.31 eV is one for the association range, consistent with r 7 × 10 13 (V cm). The accurate Eao should be systematically studied in connection with preferential clusters in CaF2 doped with very low RE concentration. The decrease of the activation energy (DEa) for low RE concentration has been usually explained by the Lidiard– Debye–Hu¨ckel (LDH) theory [29,39,40] and is given by DEa ⫺e2 k=e 1 ⫹ kR
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Fig. 14. Calculated DEa versus n 1/3 by using the Lidiard–Debye– Hu¨ckel theory for R 0.65Rm. Large and small open circles are experimental and calculated values, respectively. The straight line is the least-mean-square fit of the small circles.
with
k2 8pNe2 n= 100 ekT
33 3⫹
where R is the radius of the virtual sphere centred on Y ion in which one yttrium ion is involved [29] and N 2.45 × 10 22 cm ⫺3 the site numbers of Ca 2⫹ ions in unit volume. If trivalent yttrium ions are homogeneously dispersed in CaF2, the maximum radius (Rm) of the sphere and yttrium concentration (n) in mol% are related by Rm 3=2p1=3 dF⫺F n=100⫺1=3
34
where dF–F is one half of the lattice constant of CaF2. The experimental DEa values are plotted in Fig. 14 with large empty circles where Eao 1.31 eV was used. And the calculated DEa for T 430 K could be fitted to experimental ones and are shown with small empty circles or a solid line in Fig. 14 by choosing R 0.65Rm. And the DEa is approximated, for large k R or low yttrium concentration, by DEa n 0:012 ⫺ 0:280 n1=3
35
where we neglected the influence of doped yttrium concentration on the expansion of CaF2 cell and the dielectric constant e (6.799 [35]). From the experimental results of DEa in Fig. 14, the D–n 1/3 relation is approximately linear over the wide range from n 1/3 1 to 3.1 like that observed in BaF2:Y [29]. This fact indicates that a space charge relaxes via the same relaxation mechanism in CaF2:Y. The observed wide linear range is, however, abnormal, because the LDH theory is only valid for the intermediate RE concentration range [37]. Different from the preferential cluster formation range for n 1/3 ⬍ 1 in Fig. 6, ITC due to NN Y 3⫹ ⫺ Fi⫺ dipoles observed in Fig. 1(a) is extinguished in the intermediate range in Fig. 1(b). This evolution has been usually explained by the interaction of one NN Y 3⫹ ⫺ Fi⫺ dipole with neighbouring NN dipoles [16] or clustering of NN Y 3⫹ ⫺ Fi⫺ dipoles. In fact, many types of yttrium defects
255
were observed such as 2:2:2 clusters (dimers) studied by neutron diffraction [33] and by calculation [42], gettered 2:2:2 clusters by AC dielectric loss and calculation [36], and Y 2⫹ ions in CaF2 by ESR [41]. And isolated Y 3⫹ ions like cubic Gd 3⫹ ions [44] may possibly act as the charged defect incorporated in the crystal growth. Taking into account the conversion from the latent polarizable defects to the weakly polarized ones observed as the weak ITC bands at the rising part of the intense HT ITC band, one of the possible candidates of the defects is latent polarizable dimers constructed by two NN dipoles [36,42]. And such dimers are expected to be homogeneously dispersed in the crystal to maintain the LDH theory. Combining the reported results in the thermal sampling ITC method [29] with an equivalent electric circuit analysis, we analyzed the behaviours of HT ITC in CaF2:Y. Two important results of HT ITC were found in CaF2:Y when Teflon insulated electrodes were used. One is t sys(Tm) which is two or three times larger than t cry(Tm). The other is Tm which is strongly dependent on the capacitance of the Teflon foils and the sample crystal. Accordingly, the activation energy Ea is a reliable HT ITC parameter for HT ITC.
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