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Polaronic conduction in barium borate glasses containing iron oxide
Physica B 293 (2001) 268}275
Polaronic conduction in barium borate glasses containing iron oxide E. Mansour, G.M. El-Damrawi, Y.M. Moustafa, S. Abd El-Maksoud, H. Doweidar* Glass Research Group, Physics Department, Faculty of Science, Mansoura University, P.O. Box 83, Mansoura 35516, Egypt Received 15 December 1999; received in revised form 4 May 2000; accepted 6 July 2000
Abstract The conductivity of xFe O ) (60!x)BaO ) 40B O glasses can be described using Mott's models for the small polaron hopping and variable-range hopping processes. The change of resistivity and activation energy with composition indicates that the conduction process varies from ionic to polaronic one. The activation energy is the predominant parameter that a!ects the conductivity with composition. The calculated values of the conductivity parameters indicate that the conduction mechanism is adiabatic in nature. The change in both the density and molar volume was discussed in terms of the structural modi"cations that take place in the glass matrix upon replacing BaO by Fe O . 2001 Elsevier Science B.V. All rights reserved. Keywords: Electric conduction; Polaronic hopping; Ionic conductivity; Adiabatic mechanism; Density; Molar volume
1. Introduction Conductivity of borate glasses containing su$cient amount of a transition metal oxide, such as Fe O , is mainly carried out by polarons [1,2]. During preparation of such glasses some Fe> ions are reduced to Fe> ones. The electron transfer could be ascribed as a small polaron hopping process from Fe> to Fe> ions. The electrical properties of these glasses show a semiconducting behaviour and depend greatly on the type and
* Corresponding author. Tel.: #20-50-319702; fax: #20-5046781. E-mail address: [email protected] (H. Doweidar).
concentration of the transition metal oxide as well as on the type of the host glass matrix. Some authors [3,4] studied the conductivity of barium borate glasses and glass ceramics containing Fe O . Their studies focused on the relation be tween the conductivity and the modi"cations in the glass matrix due to the heat treatment. Owing to the composition, non-bridging oxygen ions (NBOs) were not expected to be present in these glasses. In the present work, glasses in an extended forming region, compared with the previously studied glasses [3}7], will be investigated. The aim of this work is to study the characteristic features of the polaronic hopping conduction in Fe O }BaO}B O glasses, in which the formation of non-bridging oxygen ions is expected.
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 5 5 9 - 7
E. Mansour et al. / Physica B 293 (2001) 268}275
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Table 1 Chemical composition (mol%) and parameters of the studied glasses. N is the concentration of iron ions, R is the Fe}Fe distance and D is the density Glass no.
Fe O
BaO
B O
N (10 cm\)
R (As )
D (g cm\)
G1 G2 G3 G4 G5 G5a G6 G6a G7 G7a G8
0 2.5 5 10 15 17.5 20 22.5 25 27.5 30
60 57.5 55 50 45 42.5 40 37.5 35 32.5 30
40 40 40 40 40 40 40 40 40 40 40
} 1.04 2.06 4.1 5.95 6.74 7.64 8.42 9.78 10.06 11.48
} 9.87 7.85 6.24 5.52 5.29 5.08 4.92 4.67 4.63 4.43
4.131 4.149 4.120 4.100 3.981 3.871 3.843 3.768 3.939 3.693 3.873
Glasses G5a, G6a and G7a were used to check up the density and molar volume data.
2. Experimental Glasses (Table 1) of the formula xFe O ) (60!x)BaO ) 40B O were prepared by the quenching technique. The Fe O concentration ranged between 0 and 30 mol%. Small quantity of TiO (1 mol%) was added to extend the glass formation region [8,9]. Reagent grade Fe O , BaCO , H BO and TiO were mixed together and melted in porcelain crucibles in a mu%e furnace at temperatures between 10503C and 11503C, depending on the glass composition. To assure homogeneity, the well-mixed components were added in small portions and the melt was swirled frequently. The homogenized melt was poured onto a steel plate and quickly pressed by another one to obtain disks having thickness of 1 to 2 mm. For measuring DC conductivity (p) disks of thickness about 1 mm were coated with graphite to serve as electrodes. The radius of the coated area is 5 mm. The resistance was measured using an insulation tester type TM14 (Levell Electronics Ltd.) with 10}10 ) range. As a rule, three samples of each glass were used to measure the resistance. The experimental error in determining the activation energy for conduction is estimated to be less than 0.02 eV, whereas the relative error in the conductivities is expected to be $5%. The densities were determined at room temperature using Archimedes method with xylene as an
immersing liquid. Four samples of each glass were used to determine the density (D). The density values were reproducible to $0.02 g/cm. The molar volume (< ) was calculated using the formula
< " x M /D, (1)
G G where x is the molar fraction for the component (i) G and M is its molecular weight. G Analysis of Fe>/(Fe>#Fe>) ratio was carried out by titration method with standard KMnO after complete dissolution of the glass sample employing concentrated H SO [10]. 3. Results The formation of glass in the Fe O }BaO}B O system is known to be possible only if the proportion of Fe O does not exceed 20% under the normal quenching technique [11]. Although TiO is a nucleating agent for crystallization, it was found that it inhibits the precipitation of any ferrite phase in the Fe O }BaO}B O glass matrix [8,9]. In the glasses of the present work, TiO is added to prevent the crystallization as much as possible. However, X-ray di!raction patterns for the studied glasses indicated that, there are crystalline phases in the glasses containing Fe O *25 mol%, whereas the glasses with lower Fe O contents are amorphous.
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Fig. 1 shows the temperature dependence of the DC conductivity of the studied glasses. It is observed that except the glass G1, which is free from Fe O , the glasses show non-linear behaviour and the activation energy changes with temperature. In the high-temperature region there is a linear dependence of ln p on the reciprocal of absolute temperature. In this case, the dependence can be represented by the Arrhenius equation p"p exp(!=/K¹), (2) where p is the conductivity at a temperature ¹ (in K), p is the pre-exponential factor, = is the activa tion energy and K is the Boltzmann constant. In Table 2 values of = and p are given for glasses containing Fe O . They were obtained by using
Eq. (2) and the conductivity data in the high-temperature region. The obtained activation energies are shown in Fig. 2 as a function of composition. The activation energies of the base glass and glass containing 2.5 mol% Fe O are higher than that of other glasses. There is a fast decrease in = between 2.5 and 5 mol% Fe O . Fig. 2 shows a slight de crease in = for glasses G4}G7. Further increase in Fe O concentration causes another fast decrease in =. This "gure also shows that the conductivity at a certain temperature (¹"573 K), p , has an opposite trend. The composition dependence of both the density and molar volume for the glasses G1}G6a is shown in Fig. 3. It is observed that there are two regions of change. From 0 to 10 mol% Fe O both of D and
Fig. 1. Temperature dependence of the electrical conductivity p for the glasses investigated. The lines are the "tting plots of the data.
Fig. 2. The composition dependence of both high-temperature activation energy and ln p for the studied glasses. The lines are drawn as a guide for the eye.
Table 2 Some physical parameters of the studied glasses: C is the ratio Fe>/(Fe>#Fe>), W the activation energy for conduction, W the & hopping energy, l the phonon frequency and N(E ) the density of states at Fermi level $ Glass no.
C
W (eV)
ln[p () cm)\]
= (eV) &
l
G2 G3 G4 G5 G6 G7 G8
0.508 0.462 0.232 0.113 0.133 0.107 0.090
1.26 0.96 0.85 0.78 0.76 0.70 0.41
!2.20 !6.59 !5.74 !5.62 !5.14 !5.00 !6.69
0.79 0.63 0.52 0.47 0.46 0.42 0.22
} 0.302 0.408 0.530 0.574 0.359 }
)
(10 Hz)
N(E ) (10 eV\ cm\) $ 0.066 0.168 0.203 0.240 0.262 0.394 }
E. Mansour et al. / Physica B 293 (2001) 268}275
Fig. 3. Variation in the density D and molar volume < with
the Fe O content for the glasses G1}G6a. The lines are drawn as a guide for the eye.
< are nearly constant. Between 10 and 22.5 mol%
Fe O , the density decreases while the molar vol ume increases with increasing Fe O content. 4. Discussion The deviation from linearity in the ln p versus (1/¹) dependence (Fig. 1) indicates that the activation energy is temperature-dependent. This is a characteristic feature for the small polaron hopping conduction [9,12}15]. The fast decrease in the activation energy for glasses containing up to 5 mol% Fe O , Fig. 2, may re#ect the change from ionic to polaronic conduction upon introducing Fe O into the glassy matrix. The conduction in the base glass is considered ionic due to the migration of barium ions within the matrix. This is in agreement with the results of Gang et al. [16]. They reported that the conduction mechanism changes from ionic to electronic and is considered to be mainly electronic for Fe O '5 mol% in the BaO}B O }Al O glasses containing Fe O . The sudden drop in = for Fe O '25 mol% can be attributed to the formation of some crystalline phases as indicated by X-ray analysis. The general behaviour of the activation energy versus composition was also observed for xFe O !(50!x)PbO!50B O [1] and xFe O !(100!x)+PbO!3B O , [2] glasses.
271
Fig. 4. Dependence of lnp (the natural logarithm of conduct ivity at 573 K) on the polaronic activation energy = for the glasses G3}G7. The line is the "tting plot of the data.
The opposite behaviour of the conductivity, p , with respect to the activation energy (Fig. 2) suggests some dependence of conductivity on the activation energy. Fig. 4 shows a linear decrease of ln p with increasing E for the glasses having 5)Fe O )25 mol%. The behavior observed in Fig. 4 indicates that the activation energy is the predominant factor a!ecting the conductivity. This conclusion can be obtained numerically. As an example, for glass G4 at 673 K the conductivity changes from 1.42;10\ to 8.91;10\ () cm)\ due a change of 30% in p . On the other hand, the conductivity changes from 1.42;10\ to 1.36;10\ () cm)\ due a similar change in = at the same temperature. As it is known, introducing one molecule of BaO into the B O matrix converts two BO units into two BO units [17]. This process continues up to about 33 mol% BaO where the number of BO units reaches its maximum value. NBOs start to form for BaO'33 mol% [17]. The structure of the base glass (60BaO}40B O ) is characterized by a high oxygen/boron ratio (O/B) and therefore is relatively more open. On the other hand, Fe O acts as a glass modi"er at low concentrations while at higher concentrations it acts partially as a network former [18]. The introduction of Fe O into such open matrix at the expense of BaO do not bring a great change in the density or the molar
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E. Mansour et al. / Physica B 293 (2001) 268}275
volume up to 10 mol% Fe O (Fig. 3). The intro duction of Fe O into the matrix at higher concen trations causes a remarkable increase in the molar volume and an opposite behaviour of the density for G4}G6. The molecular weight of Fe O is slightly greater than that of BaO. The increase in < may indicate that the volume of NBO sites
produced by the modi"er Fe O is greater than that produced by an equivalent quantity of BaO. It should be noticed that, for su$cient high modi"er concentration, each BaO molecule produces two NBO sites whereas there would be four NBO sites for each modi"er Fe O molecule. For glasses con taining Fe O '22.5 mol%, an irregular behav iour in both < and D was observed. This may be
due to the e!ect of crystallization, which was observed for glasses containing Fe O *25 mol%. Fig. 5 shows the Fe}Fe distance (R) versus Fe O content for the studied glasses. Assuming the transition metal ions (TMIs) being uniformly distributed within the matrix, the mean value of R can be calculated from the relation R"(1/N) ,
(3)
where N is the number of TMIs per unit volume. The value of R shows a systematic decrease with the increasing Fe O concentration. The system atic change of the hopping distance R as a function of Fe O concentration (x) is well "tted by the equation R"13.192x\ .
(4)
Fig. 5. The composition dependence of the average Fe}Fe distance R for the glasses investigated. The line is the "tting plot of the data and it veri"es Eq. (4).
This semi-empirical formula can be used to predict the hopping distance for any composition within the region investigated assuming a homogeneous matrix. In such a case there is no need for the density of that composition. The electrical conductivity due to small polaron hopping, is given by Mott [12,19] in the nonadiabatic hopping mechanism as p"lc(1!c)(e/RK¹)exp(!2aR)exp(!=/K¹). (5) In this relation l is the phonon frequency of the lattice. c is the redox ratio, Fe>/(Fe>#Fe>), the ratio between the concentration of Fe> ions to the total concentration of iron ions, e is the electronic charge, R is the average hopping distance and a is the decay parameter of the wave function of the electron in the low valence state. The term exp(!2aR) represents the electron overlap integral between sites. The activation energy is given by ="= #= /2 (¹'h /2), (6a) & " " ="= (¹(h /2), (6b) " " where = is the polaron hopping energy, = is & " the disorder energy and h is the Debye temper" ature. Using simple electrostatic principles, Austin and Mott [13] showed that = "(e/4e )[(1/r )!(1/R)] (7) & where r is the polaron radius and e is the e!ective dielectric constant. Many authors [20}23] indicated that the small polaron radius (r ) can be calculated from the relation r "(1/2)(p/6N) . (8) According to Austin and Mott [13], the polaron binding energy (= ) is given by = "e/2e r . (9) The values of e can be calculated from Eq. (9) by considering that = can be taken as twice the high-temperature activation energy = [24]. Using the values of e (Eq. (9)), R (Eq. (3)) and r (Eq. (8)), the polaron hopping energy can be calculated from Eq. (7). The data for glasses G2}G8 are given in Table 2.
E. Mansour et al. / Physica B 293 (2001) 268}275
273
When the term exp(!2aR) in Eq. (5) approaches unity, the hopping mechanism is adiabatic and mainly controlled by the activation energy. In this case, the conductivity at su$ciently high temperatures is given by p"lc(1!c)(e/RK¹)exp(!=/K¹).
(10)
Eq. (5) can be rewritten in a form like that of Eq. (2), where p "lc(1!c)(e/RK¹)exp(!2aR) or
(11a)
p /c(1!c)"E exp(!2aR). In Eq. (11b) E denotes to
(11b)
E"l(e/RK¹)
(11c)
The correlation between the quantity ln[p /c(1!c)] and R for glasses G3}G6 is shown in Fig. 6. From the slope of the straight line in Fig. 6 a was estimated to be 0.42 As \. A value of 0.75 As \ was reported for Fe O }PbO}B O glasses [1,2], which re#ects strong localized states. The greater value of a in those glasses can be attributed to the high polarizability of the Pb> ions. The linear dependence in Fig. 6 indicates that E is constant regardless of the interchange between the parameters inside it. The changes in the para-
Fig. 7. Variation of the longitudinal optical phonon frequency l (at 573 K) as a function of composition for glasses G3}G7. The line is drawn as a guide for the eye.
meters compensate each other to give a constant value for E [2]. Eq. (5) was used to calculate the values of l from the high-temperature conductivity data and the room temperature Fe}Fe distance. The calculated values of l at 573 K are given in Table 2. In Fig. 7, there is a remarkable increase in the phonon frequency for the glasses G3}G6 against Fe O con tent, then the value of l decreases. This decrease may be due to the e!ect of crystallization. According to the model given by Mott [12], at low temperatures, depending on the type of glass, the conduction would be controlled by the variable-range hopping mechanism. In this case, the conductivity is given by p"A exp(!B/¹ ),
(12a)
where A is a constant. In turn B is given by
Fig. 6. Change in ln+p /[c(1!c)], with the average Fe}Fe distance R for the glasses G3}G7. The solid line is the "tting plot of the data.
B"2.1[a/KN(E )] , (12b) $ where N(E ) is the density of energy states at the $ Fermi level. The linear dependence of lnp on 1/¹ for glasses G2}G7 (Fig. 8) indicates that Eq. (12a) can be used to describe the conductivity behaviour in the studied range of temperature. The slope B of the plots in Fig. 8 and the estimated value of a can be used in Eq. (12b) to get the value of N(E ) for $ each glass. The estimated values of N(E ) are given $ in Table 2. It can be noticed that N(E ) generally $ increases with the increase in Fe O concentration.
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E. Mansour et al. / Physica B 293 (2001) 268}275
where J is the polaron bandwidth. The condition for small polaron is J)(1/3)= . The value of J, & can be estimated according to the formula [27] J+e[N(E )] /e . $
Fig. 8. Dependence of ln p on ¹\ for glasses G2}G7. The lines are the "tting plots of the data.
Mott [26] indicated that the linear dependence between ln p and 1/¹ would appear when hopping of polarons is between nearest neighbours. This would be highly probable in the high-temperature region (¹'h /2). At low-temperatures tunnelling " to more distant sites would be favourable. In such cases, the conduction process can be described by the variable-range hopping mechanism. It is expected that all charge carriers would not be able to make tunnelling to more distant sites at the same time. This arises as a gradual decrease in ln p while decreasing the temperature. The results of the present work shows that both the models are applicable for the studied glasses in the temperature region used. This reveals that upon decreasing the temperature (within the used region) certain fraction of the charge carriers would be able to make tunnelling to sites that are near to the next ones. The validity of 1/¹ behaviour for the conductivity at relatively high temperatures has been observed earlier [2]. Much lower temperatures are required to separate these mechanisms from each other. A check whether hopping is in the adiabatic or non-adiabatic regime can be made by using Holstein's condition [25]
J
'(2K¹= /p) (hl/p) adiabatic, & ((2K¹= /p) (hl/p) non-adiabatic, & (13)
(14)
The right-hand side (RHS) in inequality (13) is calculated using the estimated values of = and & l at di!erent temperatures, while the J value is estimated using the values of N(E ). It was found $ that, the values of J satisfy the condition J)(1/3)= which is one of the conditions of small & polaron formation. The value of J increases with an increase in the Fe O content. It varies from 0.011 eV for G2 to 0.027 eV for G7. At any temperature, the value of the RHS in inequality (13) is lower than the corresponding value of J. For example at 573 K, it changes from 0.0079 eV for G2 to 0.0065 eV for G7. Similar values for J were observed for xFe O ) (50!xPbO) ) 50B O glasses [1]. From comparing the given values of the RHS and that of J, it is concluded that the hopping is adiabatic in nature. This is con"rmed by the observation that the values of exp(!2aR) for the studied glasses is much smaller than unity and cannot be ignored in the conductivity function. In addition, the conductivity is mainly controlled by the activation energy not by the pre-exponential factor (Fig. 4) and this is a characteristic feature of the adiabatic hopping mechanism [12].
5. Conclusion The electric conduction in BaO}Fe O }B O glasses has been studied between 333 and 743 K. In this region both the small polaron hopping and the variable-range hopping models are adequate to describe the DC conduction. These models are applicable for 5)Fe O )25 mol%. The electric conduction process varies from ionic to polaronic one upon increasing the Fe O content. The polar on transport is adiabatic in nature. Formation of crystalline phases for Fe O '25 mol% greatly a!ects the glass properties. The change in both the density and molar volume is correlated with the structural modi"cations that take place in the glass matrix upon replacing BaO by Fe O .
E. Mansour et al. / Physica B 293 (2001) 268}275
References [1] H. Doweidar, G.M. El-Damrawi, Y.M. Moustafa, J. Phys.: Condens. Matter 6 (1994) 8829. [2] H. Doweidar, Y.M. Moustafa, A.H. Sultan, A.A. Al-Ha"ez, Phys. Chem. Glasses 38 (1997) 239. [3] A.M. Sanad, I. Kashif, A.M. Abou-Al-Azm, A.M. Khaled, H. Farouk, Egypt. J Solids 7 (1985) 71. [4] D. Hulsenberg, O. Knauf, J. Non-Cryst. Solids 80 (1}3) (1986) 474. [5] I. Bratu, P. Oana, Monica Culea, Phys. Stat. Sol. 100 (1987) K195. [6] A.M. Sanad, I. Kashif, A.M. Abou-Al-Azm, A.M. Khaled, H. Farouk, J Mater. Sci. 21 (1986) 230. [7] A.M. Sanad, I. Kashif, A.A. El-Sharkawy, A.A. El-Saghier, H. Farouk, J Mater. Sci. 21 (1986) 3483. [8] S. Ram, K. Ram, Infrared Phys. 29 (1989) 895. [9] S. Ram, D. Chakravorty, D. Bahahur, J. Magn. Magn. Mater. 62 (1986) 221. [10] A.I. Vogel, Textbook of Quantitative Inorganic Analysis, Longman, London, 1972. [11] A. Herczog, Glass Ind. 48 (1967) 445. [12] N.F. Mott, J. Non-Cryst. Solids 1 (1968) 1. [13] I.G. Austin, N.F. Mott, Adv. Phys. 18 (1969) 41.
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[14] J. Shnakenberg, Phys. Stat. Sol. 28 (1968) 623. [15] D. Emin, T. Holstein, Ann. Phys., NY 53 (1969) 439. [16] M.J. Gang, Y.J. Kim, S.J. Moon, S.J. Chung, D.J. Cha, New Phys. 28 (1988) 605 (in Korean), cited in PA, abstract number 76547, 92(1365) (1989) 6481. [17] P.J. Bray, in: L.D. Pye, V.D. FreH chette, N.J. Kreidl (Eds.), Borate Glasses, Structure, Properties, Applications, Plenum, New York, 1978. [18] T. Nishida, T. Shiotsuki, Y. Takashima, J. Non. Cryst. Solids 43 (1981) 123. [19] N.F. Mott, Adv. Phys. 16 (1967) 49. [20] G.N. Greaves, J. Non-Cryst. Solids 11 (1973) 427. [21] M. Sayer, A. Mansing, Phys. Rev. B 6 (1972) 4629. [22] M. Sayer, A. Mansing, J.M. Roeys, G. Rosenblatt, J. Appl. Phys. 42 (1971) 2827. [23] V.N. Bogomolov, E.K. Kudinov, Y.A. Frisov, Sov. Phys.Solid State 9 (1968) 2502. [24] B.K. Chaudhuri, K. Chaudhuri, K.K. Som, J. Phys. Chem. Solids 50 (1989) 1137. [25] T. Holstein, Ann. Phys. 8 (1969) 343. [26] N.F. Mott, Philos. Mag. 19 (1969) 835. [27] N.F. Mott, E.A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd Ed., Oxford University, Oxford, 1979.
Polaronic conduction in barium borate glasses containing iron oxide E. Mansour, G.M. El-Damrawi, Y.M. Moustafa, S. Abd El-Maksoud, H. Doweidar* Glass Research Group, Physics Department, Faculty of Science, Mansoura University, P.O. Box 83, Mansoura 35516, Egypt Received 15 December 1999; received in revised form 4 May 2000; accepted 6 July 2000
Abstract The conductivity of xFe O ) (60!x)BaO ) 40B O glasses can be described using Mott's models for the small polaron hopping and variable-range hopping processes. The change of resistivity and activation energy with composition indicates that the conduction process varies from ionic to polaronic one. The activation energy is the predominant parameter that a!ects the conductivity with composition. The calculated values of the conductivity parameters indicate that the conduction mechanism is adiabatic in nature. The change in both the density and molar volume was discussed in terms of the structural modi"cations that take place in the glass matrix upon replacing BaO by Fe O . 2001 Elsevier Science B.V. All rights reserved. Keywords: Electric conduction; Polaronic hopping; Ionic conductivity; Adiabatic mechanism; Density; Molar volume
1. Introduction Conductivity of borate glasses containing su$cient amount of a transition metal oxide, such as Fe O , is mainly carried out by polarons [1,2]. During preparation of such glasses some Fe> ions are reduced to Fe> ones. The electron transfer could be ascribed as a small polaron hopping process from Fe> to Fe> ions. The electrical properties of these glasses show a semiconducting behaviour and depend greatly on the type and
* Corresponding author. Tel.: #20-50-319702; fax: #20-5046781. E-mail address: [email protected] (H. Doweidar).
concentration of the transition metal oxide as well as on the type of the host glass matrix. Some authors [3,4] studied the conductivity of barium borate glasses and glass ceramics containing Fe O . Their studies focused on the relation be tween the conductivity and the modi"cations in the glass matrix due to the heat treatment. Owing to the composition, non-bridging oxygen ions (NBOs) were not expected to be present in these glasses. In the present work, glasses in an extended forming region, compared with the previously studied glasses [3}7], will be investigated. The aim of this work is to study the characteristic features of the polaronic hopping conduction in Fe O }BaO}B O glasses, in which the formation of non-bridging oxygen ions is expected.
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 5 5 9 - 7
E. Mansour et al. / Physica B 293 (2001) 268}275
269
Table 1 Chemical composition (mol%) and parameters of the studied glasses. N is the concentration of iron ions, R is the Fe}Fe distance and D is the density Glass no.
Fe O
BaO
B O
N (10 cm\)
R (As )
D (g cm\)
G1 G2 G3 G4 G5 G5a G6 G6a G7 G7a G8
0 2.5 5 10 15 17.5 20 22.5 25 27.5 30
60 57.5 55 50 45 42.5 40 37.5 35 32.5 30
40 40 40 40 40 40 40 40 40 40 40
} 1.04 2.06 4.1 5.95 6.74 7.64 8.42 9.78 10.06 11.48
} 9.87 7.85 6.24 5.52 5.29 5.08 4.92 4.67 4.63 4.43
4.131 4.149 4.120 4.100 3.981 3.871 3.843 3.768 3.939 3.693 3.873
Glasses G5a, G6a and G7a were used to check up the density and molar volume data.
2. Experimental Glasses (Table 1) of the formula xFe O ) (60!x)BaO ) 40B O were prepared by the quenching technique. The Fe O concentration ranged between 0 and 30 mol%. Small quantity of TiO (1 mol%) was added to extend the glass formation region [8,9]. Reagent grade Fe O , BaCO , H BO and TiO were mixed together and melted in porcelain crucibles in a mu%e furnace at temperatures between 10503C and 11503C, depending on the glass composition. To assure homogeneity, the well-mixed components were added in small portions and the melt was swirled frequently. The homogenized melt was poured onto a steel plate and quickly pressed by another one to obtain disks having thickness of 1 to 2 mm. For measuring DC conductivity (p) disks of thickness about 1 mm were coated with graphite to serve as electrodes. The radius of the coated area is 5 mm. The resistance was measured using an insulation tester type TM14 (Levell Electronics Ltd.) with 10}10 ) range. As a rule, three samples of each glass were used to measure the resistance. The experimental error in determining the activation energy for conduction is estimated to be less than 0.02 eV, whereas the relative error in the conductivities is expected to be $5%. The densities were determined at room temperature using Archimedes method with xylene as an
immersing liquid. Four samples of each glass were used to determine the density (D). The density values were reproducible to $0.02 g/cm. The molar volume (< ) was calculated using the formula
< " x M /D, (1)
G G where x is the molar fraction for the component (i) G and M is its molecular weight. G Analysis of Fe>/(Fe>#Fe>) ratio was carried out by titration method with standard KMnO after complete dissolution of the glass sample employing concentrated H SO [10]. 3. Results The formation of glass in the Fe O }BaO}B O system is known to be possible only if the proportion of Fe O does not exceed 20% under the normal quenching technique [11]. Although TiO is a nucleating agent for crystallization, it was found that it inhibits the precipitation of any ferrite phase in the Fe O }BaO}B O glass matrix [8,9]. In the glasses of the present work, TiO is added to prevent the crystallization as much as possible. However, X-ray di!raction patterns for the studied glasses indicated that, there are crystalline phases in the glasses containing Fe O *25 mol%, whereas the glasses with lower Fe O contents are amorphous.
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Fig. 1 shows the temperature dependence of the DC conductivity of the studied glasses. It is observed that except the glass G1, which is free from Fe O , the glasses show non-linear behaviour and the activation energy changes with temperature. In the high-temperature region there is a linear dependence of ln p on the reciprocal of absolute temperature. In this case, the dependence can be represented by the Arrhenius equation p"p exp(!=/K¹), (2) where p is the conductivity at a temperature ¹ (in K), p is the pre-exponential factor, = is the activa tion energy and K is the Boltzmann constant. In Table 2 values of = and p are given for glasses containing Fe O . They were obtained by using
Eq. (2) and the conductivity data in the high-temperature region. The obtained activation energies are shown in Fig. 2 as a function of composition. The activation energies of the base glass and glass containing 2.5 mol% Fe O are higher than that of other glasses. There is a fast decrease in = between 2.5 and 5 mol% Fe O . Fig. 2 shows a slight de crease in = for glasses G4}G7. Further increase in Fe O concentration causes another fast decrease in =. This "gure also shows that the conductivity at a certain temperature (¹"573 K), p , has an opposite trend. The composition dependence of both the density and molar volume for the glasses G1}G6a is shown in Fig. 3. It is observed that there are two regions of change. From 0 to 10 mol% Fe O both of D and
Fig. 1. Temperature dependence of the electrical conductivity p for the glasses investigated. The lines are the "tting plots of the data.
Fig. 2. The composition dependence of both high-temperature activation energy and ln p for the studied glasses. The lines are drawn as a guide for the eye.
Table 2 Some physical parameters of the studied glasses: C is the ratio Fe>/(Fe>#Fe>), W the activation energy for conduction, W the & hopping energy, l the phonon frequency and N(E ) the density of states at Fermi level $ Glass no.
C
W (eV)
ln[p () cm)\]
= (eV) &
l
G2 G3 G4 G5 G6 G7 G8
0.508 0.462 0.232 0.113 0.133 0.107 0.090
1.26 0.96 0.85 0.78 0.76 0.70 0.41
!2.20 !6.59 !5.74 !5.62 !5.14 !5.00 !6.69
0.79 0.63 0.52 0.47 0.46 0.42 0.22
} 0.302 0.408 0.530 0.574 0.359 }
)
(10 Hz)
N(E ) (10 eV\ cm\) $ 0.066 0.168 0.203 0.240 0.262 0.394 }
E. Mansour et al. / Physica B 293 (2001) 268}275
Fig. 3. Variation in the density D and molar volume < with
the Fe O content for the glasses G1}G6a. The lines are drawn as a guide for the eye.
< are nearly constant. Between 10 and 22.5 mol%
Fe O , the density decreases while the molar vol ume increases with increasing Fe O content. 4. Discussion The deviation from linearity in the ln p versus (1/¹) dependence (Fig. 1) indicates that the activation energy is temperature-dependent. This is a characteristic feature for the small polaron hopping conduction [9,12}15]. The fast decrease in the activation energy for glasses containing up to 5 mol% Fe O , Fig. 2, may re#ect the change from ionic to polaronic conduction upon introducing Fe O into the glassy matrix. The conduction in the base glass is considered ionic due to the migration of barium ions within the matrix. This is in agreement with the results of Gang et al. [16]. They reported that the conduction mechanism changes from ionic to electronic and is considered to be mainly electronic for Fe O '5 mol% in the BaO}B O }Al O glasses containing Fe O . The sudden drop in = for Fe O '25 mol% can be attributed to the formation of some crystalline phases as indicated by X-ray analysis. The general behaviour of the activation energy versus composition was also observed for xFe O !(50!x)PbO!50B O [1] and xFe O !(100!x)+PbO!3B O , [2] glasses.
271
Fig. 4. Dependence of lnp (the natural logarithm of conduct ivity at 573 K) on the polaronic activation energy = for the glasses G3}G7. The line is the "tting plot of the data.
The opposite behaviour of the conductivity, p , with respect to the activation energy (Fig. 2) suggests some dependence of conductivity on the activation energy. Fig. 4 shows a linear decrease of ln p with increasing E for the glasses having 5)Fe O )25 mol%. The behavior observed in Fig. 4 indicates that the activation energy is the predominant factor a!ecting the conductivity. This conclusion can be obtained numerically. As an example, for glass G4 at 673 K the conductivity changes from 1.42;10\ to 8.91;10\ () cm)\ due a change of 30% in p . On the other hand, the conductivity changes from 1.42;10\ to 1.36;10\ () cm)\ due a similar change in = at the same temperature. As it is known, introducing one molecule of BaO into the B O matrix converts two BO units into two BO units [17]. This process continues up to about 33 mol% BaO where the number of BO units reaches its maximum value. NBOs start to form for BaO'33 mol% [17]. The structure of the base glass (60BaO}40B O ) is characterized by a high oxygen/boron ratio (O/B) and therefore is relatively more open. On the other hand, Fe O acts as a glass modi"er at low concentrations while at higher concentrations it acts partially as a network former [18]. The introduction of Fe O into such open matrix at the expense of BaO do not bring a great change in the density or the molar
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volume up to 10 mol% Fe O (Fig. 3). The intro duction of Fe O into the matrix at higher concen trations causes a remarkable increase in the molar volume and an opposite behaviour of the density for G4}G6. The molecular weight of Fe O is slightly greater than that of BaO. The increase in < may indicate that the volume of NBO sites
produced by the modi"er Fe O is greater than that produced by an equivalent quantity of BaO. It should be noticed that, for su$cient high modi"er concentration, each BaO molecule produces two NBO sites whereas there would be four NBO sites for each modi"er Fe O molecule. For glasses con taining Fe O '22.5 mol%, an irregular behav iour in both < and D was observed. This may be
due to the e!ect of crystallization, which was observed for glasses containing Fe O *25 mol%. Fig. 5 shows the Fe}Fe distance (R) versus Fe O content for the studied glasses. Assuming the transition metal ions (TMIs) being uniformly distributed within the matrix, the mean value of R can be calculated from the relation R"(1/N) ,
(3)
where N is the number of TMIs per unit volume. The value of R shows a systematic decrease with the increasing Fe O concentration. The system atic change of the hopping distance R as a function of Fe O concentration (x) is well "tted by the equation R"13.192x\ .
(4)
Fig. 5. The composition dependence of the average Fe}Fe distance R for the glasses investigated. The line is the "tting plot of the data and it veri"es Eq. (4).
This semi-empirical formula can be used to predict the hopping distance for any composition within the region investigated assuming a homogeneous matrix. In such a case there is no need for the density of that composition. The electrical conductivity due to small polaron hopping, is given by Mott [12,19] in the nonadiabatic hopping mechanism as p"lc(1!c)(e/RK¹)exp(!2aR)exp(!=/K¹). (5) In this relation l is the phonon frequency of the lattice. c is the redox ratio, Fe>/(Fe>#Fe>), the ratio between the concentration of Fe> ions to the total concentration of iron ions, e is the electronic charge, R is the average hopping distance and a is the decay parameter of the wave function of the electron in the low valence state. The term exp(!2aR) represents the electron overlap integral between sites. The activation energy is given by ="= #= /2 (¹'h /2), (6a) & " " ="= (¹(h /2), (6b) " " where = is the polaron hopping energy, = is & " the disorder energy and h is the Debye temper" ature. Using simple electrostatic principles, Austin and Mott [13] showed that = "(e/4e )[(1/r )!(1/R)] (7) & where r is the polaron radius and e is the e!ective dielectric constant. Many authors [20}23] indicated that the small polaron radius (r ) can be calculated from the relation r "(1/2)(p/6N) . (8) According to Austin and Mott [13], the polaron binding energy (= ) is given by = "e/2e r . (9) The values of e can be calculated from Eq. (9) by considering that = can be taken as twice the high-temperature activation energy = [24]. Using the values of e (Eq. (9)), R (Eq. (3)) and r (Eq. (8)), the polaron hopping energy can be calculated from Eq. (7). The data for glasses G2}G8 are given in Table 2.
E. Mansour et al. / Physica B 293 (2001) 268}275
273
When the term exp(!2aR) in Eq. (5) approaches unity, the hopping mechanism is adiabatic and mainly controlled by the activation energy. In this case, the conductivity at su$ciently high temperatures is given by p"lc(1!c)(e/RK¹)exp(!=/K¹).
(10)
Eq. (5) can be rewritten in a form like that of Eq. (2), where p "lc(1!c)(e/RK¹)exp(!2aR) or
(11a)
p /c(1!c)"E exp(!2aR). In Eq. (11b) E denotes to
(11b)
E"l(e/RK¹)
(11c)
The correlation between the quantity ln[p /c(1!c)] and R for glasses G3}G6 is shown in Fig. 6. From the slope of the straight line in Fig. 6 a was estimated to be 0.42 As \. A value of 0.75 As \ was reported for Fe O }PbO}B O glasses [1,2], which re#ects strong localized states. The greater value of a in those glasses can be attributed to the high polarizability of the Pb> ions. The linear dependence in Fig. 6 indicates that E is constant regardless of the interchange between the parameters inside it. The changes in the para-
Fig. 7. Variation of the longitudinal optical phonon frequency l (at 573 K) as a function of composition for glasses G3}G7. The line is drawn as a guide for the eye.
meters compensate each other to give a constant value for E [2]. Eq. (5) was used to calculate the values of l from the high-temperature conductivity data and the room temperature Fe}Fe distance. The calculated values of l at 573 K are given in Table 2. In Fig. 7, there is a remarkable increase in the phonon frequency for the glasses G3}G6 against Fe O con tent, then the value of l decreases. This decrease may be due to the e!ect of crystallization. According to the model given by Mott [12], at low temperatures, depending on the type of glass, the conduction would be controlled by the variable-range hopping mechanism. In this case, the conductivity is given by p"A exp(!B/¹ ),
(12a)
where A is a constant. In turn B is given by
Fig. 6. Change in ln+p /[c(1!c)], with the average Fe}Fe distance R for the glasses G3}G7. The solid line is the "tting plot of the data.
B"2.1[a/KN(E )] , (12b) $ where N(E ) is the density of energy states at the $ Fermi level. The linear dependence of lnp on 1/¹ for glasses G2}G7 (Fig. 8) indicates that Eq. (12a) can be used to describe the conductivity behaviour in the studied range of temperature. The slope B of the plots in Fig. 8 and the estimated value of a can be used in Eq. (12b) to get the value of N(E ) for $ each glass. The estimated values of N(E ) are given $ in Table 2. It can be noticed that N(E ) generally $ increases with the increase in Fe O concentration.
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E. Mansour et al. / Physica B 293 (2001) 268}275
where J is the polaron bandwidth. The condition for small polaron is J)(1/3)= . The value of J, & can be estimated according to the formula [27] J+e[N(E )] /e . $
Fig. 8. Dependence of ln p on ¹\ for glasses G2}G7. The lines are the "tting plots of the data.
Mott [26] indicated that the linear dependence between ln p and 1/¹ would appear when hopping of polarons is between nearest neighbours. This would be highly probable in the high-temperature region (¹'h /2). At low-temperatures tunnelling " to more distant sites would be favourable. In such cases, the conduction process can be described by the variable-range hopping mechanism. It is expected that all charge carriers would not be able to make tunnelling to more distant sites at the same time. This arises as a gradual decrease in ln p while decreasing the temperature. The results of the present work shows that both the models are applicable for the studied glasses in the temperature region used. This reveals that upon decreasing the temperature (within the used region) certain fraction of the charge carriers would be able to make tunnelling to sites that are near to the next ones. The validity of 1/¹ behaviour for the conductivity at relatively high temperatures has been observed earlier [2]. Much lower temperatures are required to separate these mechanisms from each other. A check whether hopping is in the adiabatic or non-adiabatic regime can be made by using Holstein's condition [25]
J
'(2K¹= /p) (hl/p) adiabatic, & ((2K¹= /p) (hl/p) non-adiabatic, & (13)
(14)
The right-hand side (RHS) in inequality (13) is calculated using the estimated values of = and & l at di!erent temperatures, while the J value is estimated using the values of N(E ). It was found $ that, the values of J satisfy the condition J)(1/3)= which is one of the conditions of small & polaron formation. The value of J increases with an increase in the Fe O content. It varies from 0.011 eV for G2 to 0.027 eV for G7. At any temperature, the value of the RHS in inequality (13) is lower than the corresponding value of J. For example at 573 K, it changes from 0.0079 eV for G2 to 0.0065 eV for G7. Similar values for J were observed for xFe O ) (50!xPbO) ) 50B O glasses [1]. From comparing the given values of the RHS and that of J, it is concluded that the hopping is adiabatic in nature. This is con"rmed by the observation that the values of exp(!2aR) for the studied glasses is much smaller than unity and cannot be ignored in the conductivity function. In addition, the conductivity is mainly controlled by the activation energy not by the pre-exponential factor (Fig. 4) and this is a characteristic feature of the adiabatic hopping mechanism [12].
5. Conclusion The electric conduction in BaO}Fe O }B O glasses has been studied between 333 and 743 K. In this region both the small polaron hopping and the variable-range hopping models are adequate to describe the DC conduction. These models are applicable for 5)Fe O )25 mol%. The electric conduction process varies from ionic to polaronic one upon increasing the Fe O content. The polar on transport is adiabatic in nature. Formation of crystalline phases for Fe O '25 mol% greatly a!ects the glass properties. The change in both the density and molar volume is correlated with the structural modi"cations that take place in the glass matrix upon replacing BaO by Fe O .
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