Electrical properties of semiconducting oxide glasses

Journal of Non-Crystalline Solids 32 (1979) 91-104 © North-Holland Publishing Company

ELECTRICAL PROPERTIES OF SEMICONDUCTING OXIDE GLASSES L. MURAWSKI *, C.H. CHUNG and J.D. MACKENZIE Materials Department, University of California, Los Angeles, CA 90024, USA Received 5 August 1978

Studies of the electrical conductivity of semiconducting oxide glasses are reviewed in the framework of Mott's theory for such materials. An examination of the conduction processes in semiconducting oxide glassesconfirms the applicability of the polaronic hopping model of electrical transport. Some deviation from Mott's theory are observed in phase-separated glasses. The thermal activation energy for conduction appears to be the dominating factor which controls conductivity, although in many cases the pre-exponential factor also has a great influence on conductivity. The diffusion-like conduction mechanism in a system of randomly'distributed ions is probably not applicable to glasses that exhibit some kind of order in the structure, such as clustering.

1. Introduction Oxide glasses containing transition metal ions were first reported to have semiconducting properties in 1954 [1]. Since then most studies have been on systems based on phosphates, although semiconducting oxide glasses based on other glass formers have also been made. Early work up to 1964 has been reviewed by Mackenzie [2]. More recent reviewers have treated semiconducting oxide glasses as a part of the general problem of electrical properties of non-crystalline materials [3,4] or were concerned with only the phosphates [5,6]. In this paper we will consider the theoretical and experimental relationship between various families of semiconducting oxide glasses, especially in the light of Sir Nevill Mott's contributions. Present treatments of electrical properties of semiconducting oxide glasses are based mainly on the theories of Mott [7] and Austin and Mott [8]. A general condition for semiconducting behavior is that the transition-metal ions should be capable of existing in more than one valence state, so that conduction can take place by the transfer of electrons from low to high valence states. It is not difficult to obtain transition-metal ions in two valence states for phosphate glasses containing V2Os, Fe2Os, WOs and MoOs. However, the amounts of reduced ions (V ~', Ws÷, Mo s+) are generally small unless reducing agents are introduced into the melt Or if the melting * Permanent address: Institute of Physics, Technical University of Gdansk, Poland. 91

92

L. Murawski et al. / Electrical properties of oxide glasses

process is carried out under controlled atmospheres. These reduction processe are especially needed for Cu and Ti ions in phosphate glasses [9,10]. On the other hand, Mn, Co and Ni ions in glasses exhibit different behavior. When they are present in large concentrations, almost all these ions exist in the M2. valence state rather than the higher valence states. Consequently, these glasses show very low conductivity (<10 -16 ~-1 cm-i at room temperature) [5] in contrast to the V 2 O s P2Os glasses [11] or the V2Os-TeO2 glasses [12] in which very high conductivity has been reported ("-10 -a ~2-1 cm -l at room temperature). The drift mobility calculated from conductivity is exceedingly low (<10 -4 cm 2 V -1 s-l), even for the high V2Os-containing phosphate glasses [ 13]. In this case the electron moves slowly and interacts strongly with the network. The potential energy well produced by this deformation of the network is sufficient to trap the electron on a particular ion site (i.e. the strong electron-network interaction involves a localization of charge carrier) and forms a small polaron. Moreover, because the electron is trapped on a single ion, conduction has the character of a thermal activation process. This type of conduction is also true for crystalline narrow-band and transition-metal oxides

[81. A second localization process occurs mainly in non-crystalline solids and is a consequence of the lack of long-range order. This kind of localization was first discussed by Anderson [14]. Using a tight-binding model, he has shown that all the states are localized if the ratio of the mean disorder energy potential WD between the ions to the total bandwidth approached some critical value. For 3d semiconductors Anderson's criterion for localization can be written [15] WD ~> 6(2JZ),

(1)

where J is the bandwidth related to the electron wave function overlap and Z the number of nearest neighbors. In transition-metal oxide glasses we expect the 3d bands to form localized states in the Anderson sense and therefore any polaron hopping energy WH will be increased by a disorder term WD. We will first discuss the theoretical aspects of dc conductivity and thermoelectric power and then examine the applicability of such theories to observed experimental results.

2. DC

conductivity

If a carrier remains in the vicinity of a particular atomic or ionic site (for example V s+) over a time interval longer than the typical period of vibration, the ions in the neighborhood of this excess charge will have sufficient time to assume new equilibrium positions consistent with the presence of this additional charge. These atomic displacements will generally produce a potential well for the excess carrier. If this Carrier-induced potential well is sufficiently deep the carrier may occupy a bound state, because it is unable to move without an alteration of the positions of

L. Murawski et al. /Electrical properties o f oxide glasses

93

the surrounding atoms. In this case the bound carrier and its induced lattice deformation is termed a polaron. If the potential well in which the excess carrier is conf'med is essentially located at a single atomic or ionic site, one calls the polaron a "small" polaron [16]. Since the potential well resulting from the carrier-induced displacements acts to trap the carrier itself, the carrier is often referred to as being selftrapped. Fig. 1 presents Mott's picture of an electron-hopping process between a and b ions in a lattice. Initially the electron is trapped in a potential well, as in fig. la. The smallest activation energy corresponds to the state as shown in fig. lb when thermal fluctuation ensures that the wells have the same depths. The energy necessary to produce this configuration is WH -_ l Wp = e2/4epTp ,

(2)

where ep = (l/e.. - 1/es) -1, e s and e~. are the static and high-frequency dielectric constants of the material, 3~p the polaron radius which will become clear later and Wp the small-polaron binding energy defined as the total potential energy of the electron and that of its attendant lattice distortion. A general expression for the

a

a|

b ion

ion

~

I X

h)/

t

~ x

X

Fig. 1. The polarization wells for two transition metal ions in glass during the hopph3.g process: (a) before hopping, (b) thermally activated state when electron can move, (e) after hopping. (After Mott [71.)

94

L. Murawski et al. / Electrical properties o f oxide glasses

polaron binding energy followed from small-polaron theory [ 17] is W P = ~1 ~_j~h~q, q

(3)

where N is the number of centers per unit volume, Wq the angular frequency and ~'q the coupling constant for an optical phonon with wave number q. When the optical phonon spectrum is narrow, 6% = wo, dispersion can be ignored, therefore Wp = ~hWo ,

(4)

where COois the mean phonon frequency [8]. Killias [18] pointed out that eq. (2) is only correct when R, the distance between centers, is large. When the concentration of sites is large and therefore the two polarization clouds overlap, Wrt must be dependent on the jumping distance. Mott [7] modified eq. (2) to obtain WH = ~ (e2/ep)( 1/Tp - 1/R).

(5)

It is apparent that for the polaron to be small, the polaron radius 70 should be greater than the radius of the ion on which the electron is localized, but less than the distance R separating these sites. Bogomolov et al. [19] found that for crystalline solids, "yp= ½Qr/6N) l/a ,

(6)

where N is the number of sites per unit volume. Hence the polaron should decrease in size as the number of sites increases. A typical range of value for 141u is 0.2-0.3 eV calculated from eq. (5) for semiconducting oxide glasses. As previously mentioned, in disordered systems an additional term, Wo, i.e. energy difference arising from the differences of neighbors between a and b sites, may appear in the activation energy for the hopping process. In this case, the total activation energy for the hopping process in the high-temperature region is [8] W = WH + ~ WD + W~)/16WH

(7)

or I4/= (W D + 4Wn)2/16WH .

If WD < WH then one will have W"~ WH + I WD "

(8)

An exact estimation of the value of WD is rather difficult as shown by various workers [13,20-22]. One such estimate is shown by the Miller-Abrahams theory [23] for impurity conduction in doped and compensated semiconductors. The value of WB was calculated as a thermal activation energy for a random distribution of impurities in a broad-band semiconductor, and WD = (e2/es R) K ,

(9)

L. Murawski et al. / Electrical properties of oxide glasses

95

where R is the average distance between transition-metal ions, and K is a constant of order ~0.3 which depends on the compensation and is tabulated by Miller and Abrahams [23]. A typical value of WD calculated from eq. (9) is ~0.1 eV [21] for V2Os-P~Os glasses. Fig. 1 shows that transfer of a small polaron is achieved only when the energy of the bound electron on the occupied site coincides with the loca' electronic energy level on a neighboring unoccupied site. This can only be achievec by considerable lattice distortion. Transport resulting from these transitions is a multiphonon hopping process activated by optical modes which is dominant at high temperatures. The basic expression which related the drift mobility/a to the rate of hopping P and to the site separation R, is # = (eR~/~r) e .

(10)

Here, P may be written as the product of two terms [24] : P = (probability of coincidence) × (probability of transfer when the coincidence occurs), i.e. P = (Wo/2r0 exp(-W/kT) X p .

(11)

The first probability in eq. (11) is the product of the predominant phonon frequency 600 and the Boltzmann factor involving the minimum energy for coincidence. As far as factor p is concerned, two different cases should be considered. The first is the so-called "adiabatic regime" where the electron can always follow the lattice motion. In this case time duration of coincident events is long compared with the time it takes an electron to transfer between coincident sites, and therefore one can set p = 1. On the other hand, when the time required for an electron to hop is large compared with the duration of a coincident event, an electron will not always follow the lattice motion and miss many coincident events before making a hop. Then p < < 1 and p is given by

p = 2lr/hWo(rr/4WrlkT)a/2J 2 .

(12)

This equation involves the electronic transfer integral J, which is a measure of the wave-function overlap of the neighboring sites. In second case, a "non-adiabatic regime" occurs if J < hwo (i.e. the predominant phonon energy). From eqs. ( 1 0 ) (12) the mobility becomes

la

=

-eR2 - - - l [I _ _7r \1/2 | j2 ( -~T ) k T h ~gWHkT] exp.

(13)

Normally/a is dominated by the exponential term. However, in a temperature range in which Wp is of the order of kT, the pre-exponential term may influence the temperature dependence. Eq. (13) is identical to the high-temperature limit of Holstein's more general theory [ 17]. A general formula for electrical conductivity of semiconducting transition metal

96

L. Murawski et al. / Electrical properties o f oxide glasses

oxide glasses was proposed by Mott [7,8] where the conductivity is given by e =

vphe2C(1- C) exp(-2otR) exp ( - k-~), kTR

(14)

where Uph is a phonon frequency, a the rate of the wave-function decay, C the ratio of ion concentration in the low valency state to the total concentration of transition metal ions and R the average hopping distance. Eq. (14) can be compared to the common Arrhenius equation o = Oo exp(-W/kT).

(15)

In many ways, the Mott expression (14) is similar to eq. (13) for the hopping of polarons since o = enla,

(I 6)

and the term exp(-2aR) in eq. (14) describes the overlap of the wave functions of neighboring hopping sites. It therefore corresponds to the j2 term in eq. (13). Similarly, eq. (14) would predict a non-adiabatic regime if Uph exp(-2aR) < Wo/2rr.

(17)

A maximum conductivity would be expected from eq. (14) when C = ½. As the temperature is lowered the multiphonon processes are frozen out and the high-temperature hopping activation energy is expected to drop continuously from Wr~+ ½WD to ~WD. In the low-temperature range T < 0[4, charge carrier tansport should be an acoustical phonon-assisted hopping process having an activation energy ~WD, whereas in the high-temperature range T > O [ 2 an optical multiphonon process takes place and the activation energy should be Wn + ½WD. Here 0 is the Debye temperature defined by kO = h w o . A detailed theory was given by Schnakenberg [25] and experimental evidence was discussed by Greaves [20], Sayer et al. [21] and Linsley et al. [11] for P2Os-V2Os glasses. Mott [7] has pointed out that at very low temperatures the observed value for Wo should approach zero because the most probable jump will not be to nearest neighbors but to more distant sites where the energy difference is small. The temperature dependence of conductivity under this condition is given by In o = A - B I T 1/4

(18)

where A and B are constant and B is given by B = 2.1 [t~a/kN(EF)] 1/4 = 2.4 [WD(otR)a/k] v 4 , where N ( E F ) is a density of states at the Fermi level. Thus, one would expect a continuous curvature on the plots of log o versus (l/T) over a wide temperature range.

L. Murawski et al. / Electrical properties of oxide glasses

97

3. Thermoelectric power The thermopower S of semiconducting oxide glasses is of interest because of the information it yields on WD due to the random fields. Mott [7] pointed out that for conduction in a material having an impurity bandwidth (Wo in our case) greater than kT, whether transport is by hopping or not, the "metallic" type formula for thermopower can be applied, i.e. S-

(19)

lr2 k2--T( d l n ° ] 3 e \ dE ] E=EF "

Substituting the temperature dependence for conductivity [eq. (14)] into eq. (19), one would obtain n2kFd(ln Oo) S =-~-eL clE

d~.] ,

when WD > k T

(20)

and should expect a temperature dependence of thermopower. However, when Wo < kT, the thermopower obeys the Heikes and Ure [26] relationship and is temperature independent, S = k/e [In(C/1 - C) + a'] ,

(21)

where a' is a constant representing the change in the entropy of the lattice due to the presence of an electron on a transition metal site. Usually, this constant is small [27] and can be negligible. As a result, the Seebeck coefficient depends only on the ratio of high to low valency, i.e. S = k/e [ln(C/1 - C)] .

(22)

Here, the density of carriers is C (i.e. V~ , Fe 2+) and the density of states available to these carriers is (1 - 6 ' ) (i.e. V s+, Fe3+). The temperature independence of thermopower (at room temperature and above) denotes that all available carriers are mobile, and one can therefore take the V~ concentration in vanadium-containing glasses, for example, as the free carrier concentration.

4. Discussion of experimental data The experimental data in almost all publications are discussed in terms of the Mott theory [7,8] of electrical conductivity in transition metal oxide glasses. Eq (14) is usually applied for the analysis of dc conduct!vity. It is interesting to compare the experimental data with the predictions of this theory, namely eq. (14), in various systems. Two terms selected for comparison are the pre-exponential factor o0 of eq. (15) and the activation energy. In the pre-exponential factor, the ratio C, the tunnelling term exp(-2oa~) and the average hopping distance R are particularly important for the value of conductivity.

98

L. Murawski et al. / Electrical properties of oxide glasses

4.1. The pre-exponential term Oo 4.1.1. The conductivity dependence on the ratio C The conduction mechanism in semiconducting glass is diffusion-like in nature and the model is based on the random distribution of ions in glass. Therefore, the presence of heterogenity would cause the conductivity to deviate from the prediction of theory. The fact that crystallization can increase conductivity is well known and has been investigated by many authors [28-30]. Effects from glass-glass phase separation has been mentioned, but experimental evidence is not abundant. For instance, Kinser and Wilson [6] have studied the electrical properties and the corresponding microstructures of vanadium phosphate glasses and suggested that the observed conductivity maximum at C < 0.5 is a consequence of microstructural segregation [6,31]. Another example is the work of Anderson and MacCrone [32] who pointed out that in iron silicate glasses the majority of the iron ions are situated in relatively well-ordered clusters containing various numbers of iron ions. Most transition-metal-oxide-containing phosphate glasses are considered to be homogeneous and exhibit good agreement with the Mott equation [eq. (14)]. However, the maximum conductivity at C ~ 0.5 only occurs in iron phosphate glasses among all the glasses which have been throughly investigated, i.e. the V2OsP2Os [11], CuO-P2Os [9] and FeO-P2Os [33,34] systems (fig. 2). In CuO-P2Os glasses conductivity increased with ratio C = Cu÷/Cutotal and no maximum has been observed. Tsuchiya and Moriya [9] suggested that this effect is a a consequence of ionic conduction of Cu + ions. In other words, CuO-P2Os glasses exhibit a "mixed conduction" phenomenon in which ionic conduction as well as electronic conduction occur in the glass. In vanadium phosphate glasses the maximum occurs at a V4+/Vtotal ratio between 0.1 and 13.2 (fig. 2). This maximum corresponds to a weak minimum in the activation energy [11], Similar behavior was observed in V2Os-TeO2 glasses. The conductivity is at a maximum when V'~/ VtotaI ~ 0.2 [12]. Various explanations have been proposed for this deviation from C = 0.5. Some explanations are based on the ideas that a fraction of the V 4÷ ions are firmly trapped in complexes [11]. Others have stressed the importance of polaron-polaron interaction at high carrier concentrations [35]. Sayer and Mansingh [5] have proposed that correlation effect due to short-range Coulomb repulsion will modify C in eq. (14) to C(1 - C)n, where n is the number of sites surrounding the polaron at which strong interaction occurs. Despite all these suggestions, it seems that the effects from phase separation have not been adequately treated [6]. In fact homogeneous barium borate glasses containing V2Os up to 35 mol% were found to show a maximum conductivity at C = 0.45 [36] (fig. 2). A very different behavior has been observed for barium borosilicate glasses (BaO-B2Oa-SiO2) containing titanium ions up to 12 mol%. The electrical conductivities of these glasses are lower than 10 -~2 ~2-1 cm -~ at room temperature. As is shown in fig. 2, the dc conductivity does not follow the C(1 - C ) dependence [37]. The ac activation energies for glasses containing more than 6% of Ti ions were found to be constant and indepen-

99

L. Murawski et al. / Electrical properties o f oxide glasses • v 2 o s - ~ o s ~111 • FeO -- P 2 0 5 [ 3 3 , 3 4 1 O v 2 O B - - B 2 0 3 - BaO [ 3 6 ] c.o -~o s (s) 1-1 | B a O - S i O 2 - 1 1 2 0 3 ) - - T i O 2 ~37)

/

/~ -8

'

o/

--9

//

--10 'E1~

¢

0 0 --11 .J

o -a

-12

-e

O •

-9

0

-13

A

I

I G2

_ I 0.4 C

I 0.6

[ 0.8

1.0

-14

( Mreduced // Mtota I )

Fig. 2. The electrical conductivity dependence on the ratio C in various glasses.

dent of the ratio C. These discrepancies suggested that the diffusion-like conduction mechanism in the system of random distribution ion sites is inappropriate. Anderson and MacCrone [32] proposed a model to explain their observation by postulating that the charge carriers move along paths at relatively high conductive chains of transition metal ions.

100

L. Murawski et al. / Electrical properties of oxide glasses

4.1.2. The tunnelling term: e x p ( - 2 a R ) The importance of the tunnelling term to the conductivity is not as obvious as that of the C term and the activation energy. Sayer and Mansingh [5] have shown that for a series of phosphate glasses containing different transition metal ions, a semilogarithmic plot of the conductivity measured at a temperature of 500 K versus the high-temperature activation energy gives a straight line with a slope corresponding to a measurement temperature of 530 K. They concluded that the pre-exponential term of eq. (14) inclusive of exp(-2aR) is virtually constant for all phosphate glasses containing different 3d transition metal ions. However, only one single point for each glass was considered with the exception of vanadium. We have now utilized all published data and plotted log o as a function of W in fig. 3. The values of temperature within the parentheses are those arbitrarily chosen to obtain the conductivity. The open values correspond to the slopes of fig. 3. Apparently, good agreement is only obtained for V2Os-P20 s and WO3-P2Os glasses. This indicates that for the other systems, the (kT) -1 values are 1.6-3.5 times higher than the experimental values if the effect of the exp(-2aR) term is ignored. Evidently, in V2OsTeO2, FeO-P2Os, MoO3-P2Os, TiO2-P2Os and V2Os-B203-BaO systems the tunnelling term should not be a constant and therefore the hopping of a small polaron in these systems should exhibit a non-adiabatic character. On the other hand, V2Os-P2Os and WO3-P2Os glasses might follow the adiabatic approximation. It should be pointed out that the conductivity is also affected by the carrier concentrations that can vary in different glasses, and this would affect the pre-exponential term, although this effect might not be large. Attempts have been made to calculate the tunnelling term, but controversies exist. For the determination of a most authors applied eq. (14) by calculating R and C values from the composition, based on the assumption that the site distribution is random and assuming a value ~'ph of 1011-1013 s-1. It is also possible, as pointed out by Greaves [20] to calculate a from small-polaron theory. Experimentally, a-values can be obtaine~l from log Oo =f(R). Murawski and Gzowski [39] have shown that log o0 is a linear function of R in iron phosphate glasses. This is consistent with the results obtained by Hirashima and Yoshida [42] for different systems of iron-containing glasses. Table 1 shows the a-values derived from this method. If the variable-range hopping is observable, the a-value can be estimated from low-temperature dependence of conductivity. In this case, the density of states N(EF) is unknown and should be evaluated. In all cases, as shown in table 4, the a-values are in the range 0.4-4.8, -1 . 4.2. Activation energy

One difficult problem to solve is the separation of the observed activation energy W into a polaron term I¢n and a disorder term WD. Attempts have been made to solve this problem. The evaluation of Wo from the Miller-Abrahams equation [eq. (9)] gives WD < 0.1 eV [21] and this value is consistent with the low-temperature

L. Murawski et al. / Electrical properties o f oxide glasses I 0

101

0 v=os-p2os (s) ~ V2OS.To02 112)

O~

O wo3- pzos ~3s)

~

•

IFeO . P~d[~ 130.311~1

0 TiO2-P20S (41)

I!

0~.o.

~)

v2%- ~2o3-1.o (361

•

•

-7

0 .J

\ W

i

(evJ

Fig. 3. The relationship between a logarithm of conductivity and activation energy in various glasses. Experimental temperatures are indicated in parentheses. The open value corresponds to the slope.

102

L. Murawski et aL / Electrical properties o f oxide glasses

Table 1 Class composition (mol%)

BI a) (eV)

V2Os-P2Os (88-49 V2Os) V2Os-P2Os (80-60 V20 s) V2Os-TeO2 (50-10 V2Os) VzOs-BzOa-BaO (37-15 V20 s) WOa-P2Os (78-65 WO3) MoOa -P2Os (85-60 MOO3) TiOz-PzOs (71-66 TiO2) FeO-P20 s (55-15 FeO) FezOs-P2Os-PbO FezOa -SiO2 -PbO Fe20 a -B20 a-PbO (15-5 Fe203) Fe203-B2Oa -BaO (20-5 Fe203)

0.29-0.42

B/D (eV) <0.1 b,c)

7p (A)

~ (A-1 )

Hopping regime (ref.)

2.1

-

a d) [5,211

0.31-0.36

0.36-0.43

2.6-2.9

2.9-4.0

b e) [20]

0.25-0.34

0.02 0

_

0.97

b 12]

0.52-1.33

<0.1 c)

2

0.89-0.45

b 361

0.29-0.35

0.1 b)

1.2-1.16

2.8-2.4

a 38]

0.5 -0.69

-

1-2

0.45-0.8

b 40]

0.48-0.54

0.05 b)

1.7

-

a 411

0.58-0.76

0.44-0.8

1.9-3.1

1.5

b 22,391

0.61-0.95 0.63-0.93 0.70-0.8

-

-

0.5 -1

b I42]

0.8-0.97

-

2.1-3.2

0.4-0.5

b [43]

a) High-temperature experimental value. b) Low-temperature BI. c) Calculated from Miller-Abrahams [23], eq. (9). d) a - Adiabatic regime e) b - Non-adiabatic regime. f) From thermopower. (<100 K) activation energy in V2Os-P2Os [21], WO3-P2Os [38] and TiO2-P~Os [41] glasses (table 1). It is also possible to calculate WD from the Mott T - U 4 lowtemperature dependence equation [eq. (18)]. But disorder energy calculated from this method is higher than low-temperature activation energy in vanadate [13] and tungsten glasses [38]. Here, Wrt can be calculated from small-polaron theory, i.e. eq. (5). Usually ep --e** = n 2 and polaron radius is calculated from the Bogomolov et al. formula [eq. (6)] developed for crystals. Good agreements have been observed for V2Os-P2Os, WO3-P2Os and TiO2-P2Os glasses. In iron-containing glasses the observed activation energy is very large. The disorder energy obtained from ~Wt) = W - WH = 0.22---0.4 eV [22] [ W . from eq. (5)] is much higher than the estimated MillerAbrahams disorder term. This discrepancy has been pointed out by Austin [15], i.e. that an additional term AU describing structural differences between transition

L. Murawski et al. / Electrical properties o f oxide glasses

10 3

metal ions should appear in the activation energy: I4/= WH + ½ WD + AU. It is likely that in V:Os-B203-BaO glasses [36] the AU term should make a large contribution to the observed activation energy. The most useful method to evaluate I¢o is from the temperature dependence of thermoelectric power. This method has been used in V:Os-TeO2 glasses [12]. From the low-temperature thermopower data the disorder energy is estimated to be ~0.02 eV. Unfortunately, this method has not been applied to other systems owing to the difficulty of measuring thermoelectric power in the low-temperature range. Finally, as is seen from table l, the requirements of applying polaron theory, namely a-1 < ~,p < R, are fulfilled in almost all glassy systems. Also, the theoretical discussion, by many authors on the application of small-polaron theory has shown that the small-polaron coupling constant in semiconducting oxide glasses is very high (~"> 10).

5. Condu~on An examination of the conduction processes in semiconducting oxide glasses suggests that the polaron model is generally applicable. Mott's equation [eq. (14)] agrees with experimental data in many cases. Particularly good agreement is observed for homogeneous glasses. The effect of phase separation probably has a great influence on the observed dependence of conductivity on C. Some discrepancies do exist in the evaluation of the tunnelling term exp(-2aR). The diffusion-like conduction mechanism in a system of randomly distributed ions is inappropriate in glasses that exhibit some kind of order in the structure. In this case the conduction path model proposed by Anderson and MacCrone [32] appears to be satisfactory. Acknowledgment

J.D. Mackenzie would like to thank the Directorate of Chemical Sciences, Air Force Office of Scientific Research, for their continuing support under Grant No. 75-2764. References [1] E.P. Denton, H. Rawson and J.E. Stanworth, Nature 173 (1954) 1030. [2] J.D. Mackenzie, in: Modern Aspects of the Vitreous State, Vol. 3, ed. J.D. Mackenzie (Butterworths, London, 1964) p. 126.

104 [3] [4] [5] [6]

L. Murawski et al. / Electrical properties of oxide glasses

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