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 elec-
trical 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 semi-
conducting 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 Macken-
zie [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 semicon-
ducting 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 con-
dition for semiconducting behavior is that the transition-metal ions should be capa-
ble 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 cal-
culated 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 car-
rier) 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 dis-
cussed 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 semiconduc-
tors 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 exam-
ple 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 deforma-
tion 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 displace-
ments acts to trap the carrier itself, the carrier is often referred to as being self-
trapped. 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 neces-
sary 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 ion b ion

a| ~ I
X

h)/ t ~
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 crystal-
line 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 fre-
quency 600 and the Boltzmann factor involving the minimum energy for coinci-
dence. 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 there-
fore 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

=
-eR2
- - - l [I _ _7r \1/2
| j2 ( -~T ) (13)
la k T h ~gWHkT] exp- .
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 tem-
perature dependence. Eq. (13) is identical to the high-temperature limit of Hol-
stein'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

vphe2C(1- C) exp(-2otR) exp ( - k-~), (14)

e = kTR

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 transi-
tion 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). Simi-
larly, 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 tem-
perature is lowered the multiphonon processes are frozen out and the high-temper-
ature 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 con-
tinuous 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.
lr2 k2--T( d l n ° ] (19)
S- 3 e \ dE ] E=EF "
Substituting the temperature dependence for conductivity [eq. (14)] into eq. (19),
one would obtain
n2kFd(ln Oo) d~.]
S =-~-eL clE , 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 tempera-
ture 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 ther-
mopower (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 com-
pare 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 predic-
tion 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 corre-
sponding microstructures of vanadium phosphate glasses and suggested that the ob-
served conductivity maximum at C < 0.5 is a consequence of microstructural segre-
gation [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 V2Os-
P2Os [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 maxi-
mum occurs at a V4+/Vtotal ratio between 0.1 and 13.2 (fig. 2). This maximum cor-
responds 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 po-
laron-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 conductiv-
ity 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-
 L. Murawski et al. / Electrical properties o f oxide glasses 99

• 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
o 0
-a --11 .J

-12

-e O A -13

• I

-9
0
I
G2
_ I
0.4
I
0.6
[
0.8 1.0
-14
C ( 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. Ander-
son and MacCrone [32] proposed a model to explain their observation by postu-
lating 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 correspond-
ing to a measurement temperature of 530 K. They concluded that the pre-exponen-
tial 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 tem-
perature within the parentheses are those arbitrarily chosen to obtain the conduc-
tivity. The open values correspond to the slopes of fig. 3. Apparently, good agree-
ment 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 experi-
mental values if the effect of the exp(-2aR) term is ignored. Evidently, in V2Os-
TeO2, 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 po-
laron 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 concen-
trations 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 distribu-
tion 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. Experimen-
tally, 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 101

I
0
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

\ i
W (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 BI a) B/D 7p ~ Hopping

(mol%) (eV) (eV) (A) (A-1 ) regime (ref.)

V2Os-P2Os 0.29-0.42 <0.1 b,c) 2.1 - a d) [5,211

(88-49 V2Os)
V2Os-P2Os 0.31-0.36 0.36-0.43 2.6-2.9 2.9-4.0 b e) [20]
(80-60 V20 s)
V2Os-TeO2 0.25-0.34 0.02 0 _ 0.97 b 12]
(50-10 V2Os)
VzOs-BzOa-BaO 0.52-1.33 <0.1 c) 2 0.89-0.45 b 361
(37-15 V20 s)
WOa-P2Os 0.29-0.35 0.1 b) 1.2-1.16 2.8-2.4 a 38]
(78-65 WO3)
MoOa -P2Os 0.5 -0.69 - 1-2 0.45-0.8 b 40]
(85-60 MOO3)
TiOz-PzOs 0.48-0.54 0.05 b) 1.7 - a 411
(71-66 TiO2)
FeO-P20 s 0.58-0.76 0.44-0.8 1.9-3.1 1.5 b 22,391
(55-15 FeO)
FezOs-P2Os-PbO 0.61-0.95 -
FezOa -SiO2 -PbO 0.63-0.93 - - 0.5 -1 b I42]
Fe20 a -B20 a-PbO 0.70-0.8 -
(15-5 Fe203)
Fe203-B2Oa -BaO 0.8-0.97 - 2.1-3.2 0.4-0.5 b [43]
(20-5 Fe203)

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 low-
temperature 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 activa-
tion 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 Miller-
Abrahams 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 ob-
served 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
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104 L. Murawski et al. / Electrical properties of oxide glasses

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