On the Conduction Mechanism of Silicate Glass Doped by Oxide Compounds of Ruthenium (Thick Film Resistors)

doi:10.4236/wjcmp.2011.12004

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World Journal of Condensed Matter Physics, 2011, 1, 19-23

doi:10.4236/wjcmp.2011.12004 Published Online May 2011 (http://www.SciRP.org/journal/wjcmp)

On the Conduction Mechanism of Silicate Glass

Doped by Oxide Compounds of Ruthenium (Thick

Film Resistors)

—Diffusion and Percolation Levels

Gulmurza Abdurakhmano v

The Institute of Power Engineering and Automation of the Uzbek Academy of Sciences, Tashkent, Uzbekistan.

Email: [email protected]

Received March 15th, 2011; revised March 23rd, 2011; accepted March 25th, 2011.

ABSTRACT

The results of the investigation of conduction mechanism of silicate glass doped by oxide compounds of ruthenium

(thick film resistor) are reported. The formation of diffusion zones in the softened glass during firing process of the

mixture of the glass and the dopant powders is considered. As the result the doping glass becomes conductive. These

diffusion zones have higher conductivity and act as percolation levels fo r the free charge carriers. The effect of te mper-

ature and duration of firing process on the conductivity of doped glass is considered. Experimental results are in a

good agreement with the model.

Keywords: Lead-Silicate Glass, Thick Film Resistors , Percola tion Levels, Doping, Conductivity, Firing Conditions

1. Introduction

The silicate glass doped by oxide compounds of ruthe-

nium (DSG) and as a result becoming an electronic con-

ductor is a functional material in thick film resistors [1],

sensors [2] and electric heaters [3]. DSG is widely

known as thick film resistors (TFR) but this name does

not characterize DSG as a material. Despite the wide use

of DSG, the mechanism of electrical conduction is not

well understood yet.

For example variable range hopping, tunneling

through potential barrier, thermal activation, effective

medium approach and combinations of them have been

exploited [4-8] to explain the temperature dependence of

the DSG resistivity

( )

, schematically shown in Fig-

ure 1. Unfortunately these models can describe

( )

the narrow range of low temperatures (region I in Figure

1) only.

The quadratic temperature dependenc e

min

() ()m

TATT

ρρ

=−+

(1)

is observed often in DSG in the region II, where m

and min

Tare the resistivity and the temperature at the

min imu m.

The quadratic dependence (1) and region III we have

investigated here [9] do not correspond to any known

models of the electrical conductivity of DSG.

There are quite a few questions which can not be un-

derstood in the framework of the above-mentioned mod-

els: how size and conductivity of the dopant particles

affect the conductivity of DSG, the mechanism by which

it is affected by temperature f

T and the duration

the firing on the room temperature value 0

and slope

of the

()

, why the glass composition affects the

conduction of the DSG, why the percolation threshold on

the

( )

shift or disappearance for glass of various

content and etc. The

( )

VCV V+= here is the vo-

lume fraction of the dopant in DSG or the doping level,

and g

- volumes of the dopant and the glass accor-

dingly, 0dg

VVV

=+ is the total volume of the specimen.

We examine these questions while ignoring th e role of

the glass in conductance of the DSG. In fact at standard

firing conditions (firing temperature f

1123K

≈

) the

lead-silicate glass softens and becomes very aggressive

substance, enable to solve appreciable amounts of oxides

of many metals (units or tens of %) [10]. So long as fir-

On the Conduction Mechanism of Silicate Glass Doped by Oxide Compounds of Ruthenium (Thick Film Resistors)

300

500

1000

T, K



III

Figure 1. Resistivity vs temperature for DSG (schematical-

ly). Regi ons I , II and I II cor r e spond to low , middle and hi gh

temperature.

ing duration of DSG is limited (usually

10min

≈

) and

mechanical mixing is not affecting the distribution of

solved atoms of the dopant in the glass, it should be dif-

fusive.

In this paper we will consider a possible effect of dif-

fusion of dopant atoms at f

T in the glass on the elec-

trical conduction of the DSG.

The production of specimens is a standard procedure

for thick film resistors and is described in many papers

and books (see, for example, [2]). The glass composi-

tions we us ed are as follows (weight %):

Glass1 SiO2 31; PbO 67; MnO2 2;

Glass2 SiO2 29; PbO 67; BaO 4.

The DSG specimens compositions are as follows

(weight %):

1) Glass1 80; Pb2Ru2O6 20;

2) Glass1 90; RuO2 10;

3) Glass2 90; RuO2 10;

4) Glass2 80; RuO2 20.

Volume fraction of dopant C is less than theoretical

percolation threshold (critical value Cc) in all cases.

2. Diffusion of Dopant Atoms into the Glass

and Formation of Percolation Levels

It should be noted that the percolation theory is consi-

dered as most common model of conduction mechanism

of the DSG [11-13]. In this theory transport of charge

takes place along the contiguous chains of intimately

contacted dopant particles (infinite cluster) and the resis-

tivity of DSG depends on the dopant volume fraction C

( )

C CC

ρρ

−

= −

(2)

where 0

is material-dependent prefactor,

is the

percolation critical volume fraction of the dopant below

which

goes to infinity,

is the dc transport critical

exponent. According to the standard theory of transport

percolation [12], C

C≈ 0.12 – 0.39 and 01.6 2tt ÷ is

the universal value for three-dimensional disordered

composites.

The expression (2) predicts percolation threshold of

( )

at C

≈, while in reality the value of

might

be as high as 7 for various compositions of glass and

doping conditions [14]. Additionally, the value of C

C is

essentially lower than the theoretical estimation in many

cases or percolation threshold disappears .

The reason for this discrepancy between the experi-

mental results and the theory of percolation is the as-

sumption that the volume fraction

of the dopant in

the DSG (after firing) is known a priori (often it equals

to the initial volume fraction of them). In fact the interac-

tion of dopant and the glass is possible during the firing

process as well as the mutual diffusion of the glass and

dopant atoms. As the result one does not have informa-

tion about C in the final stage of technological process

and about the value of the resistivity of doped region s of

the glass. So we should consider that the glass resistivity

(in the diffusion zone) is reduced by many orders of

magnitude due to diffusion (as in crystalline semicon-

ductors) and infinite cluster consists of closed or over-

lapping diffusion zones which have been formed around

the each dopant particle (Figure 2).

Let us to consider [15] that the diffusion zone is

spherical because the glass is isotropic. This diffusion

zone has the volume

(3)

There

R rl= +is the radius of the diffusion zone,

is the main radius of the dopant particles, and the dif-

fusion length is

exp 2

lDD kT

ττ



= =−





. (4)

There

is the diffusion coefficient,

( )

D DT= →∞

is the activation energy of diffusion, and

is the

Boltzmann constant. Inserting

Figure 2. Formation of diffusion zone around the dopant

particle of the radius r. There r + ld is the radius of diffusion

zone, ld is the diffusion length. L is the main distance be-

tween the nei gh bor dopant particles in the DSG.

rl+

On the Conduction Mechanism of Silicate Glass Doped by Oxide Compounds of Ruthenium (Thick Film Resistors)

( )

ddg

C VVV=+ and

( )

CVVV=+ into (2) and

taking into account (3) and (4) gives

( )

ln /

1ln( ).

VVV r

ρρ

−





+ −=









−+

(5)

Expression (5) for

()T

of DSG is in a good

agreement with experiment (Figure 3) for various

glasses and dopants and makes it possible to evaluate0

Eand

from the experimental data. Here 0

is the

fitting parameter and corresponds to average resistivity

of the diffusion zone, which is unknown and is usually

no uniform. The value of the diffusion length d

l≈0.22 –

1600 μm, evaluated in the same manner for the standard

firing conditions is essentially higher than the main dis-

tance between dopant particles

2 211

Lr r





ππ

== +−











(6)

From (6) we have

0.16L

μm for main diameter

2 0.1r≈

μm of dopant particles and volume fraction

0.16C≈

, so diffusion zones are heavily overlapped and

the whole volume of the specimen is uniformly filled by

infinite conductive cluster, as shown schematically in

Figure 4. There d

and g

are the specific weight of do-

pant and glass accordingly, m

Cis the mass fraction of

the dopant.

(a)

(b)

Figure 3. R(Tf) for DSG mentioned above (see Introduction).

Solid lines are fitting of (5) to experimental data by the least

squires me t hod .

Figure 4. Generation of percolation levels due to diffusion

(schematically). The bold curve is the infinite cluster. Gray

circles represent diffusion zones. Tf in 3 is higher than in 1.

const.

Expression (5) shows good agreement with the expe-

rimental data for DSG of other compositions [16] as well

(Figure 5).

One can also derive the effect of firing duration

the resistivity

of DSG from (2) – (4):

( )

V Vr

τ ρρ



= +−









(7)

This expression is compared with the experiment in

Figure 6.

We have simulated the effect of doping level and size

of dopant particles on the

of DSG. The simulation is

based on Fick’s equation [17] and expression (5). It is

assumed that 1) diffusion from neighbor particles of the

dopant into the glass interlayer is uniform; 2) value of

is predetermined by minimum of concentration N of

dopant atoms at the middle point being at distance L/2

from each of them, i.e.,

( )

12N xL

−

∝=

The result of the simulation of

( )

ρτ

for two sizes of

dopant particles is shown in Figure 7. It is clear on one

hand that the size of dopant particles essentially affects

on the conductivity of DSG. On the other hand

( )

small

is stronger in our simulation than in the usual

model of diffusion

( )

expRc

ττ

∝−

(the last is

shown in Figure 7 as dotted line).

Resistivity of DSG is strongly affected by the dispersiv-

ity of the dopant powder [16] – reduction of the di-ameter

of the particles from 4.5 μm up to 1.6 μm decreases

On the Conduction Mechanism of Silicate Glass Doped by Oxide Compounds of Ruthenium (Thick Film Resistors)

8.0

8.5

9.0

9.5

1.5

1.0

0.5

0.0

104











Figure 5. R(Tf), recalculated data from [16] in accordance

with (5). The solid line is fitting of (5) with R0 = 10 Ohm, t =

1.7, Vc = 0.16.

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15



min



kOhm



Figure 6. R(τ) for DSG at Tf , K: 1073 (1, 2) and 1123 (3, 4).

Temperature increasing rate is higher for data 1 and 2,

than for data 3 and 4. Points are recalculated from [16] in

accordance with (7). DSG composition isn’t varied.

0.5

1.0

1.5

2.0

2.5

3.0



,ar b.unit







Figure 7. The result of simulation

( )

of DSG for r, μm:

0.5 (a); 1 (b).

≈5 μm. The function

( )

expab c

+−

(dotted li ne) have been fitte d to curve a in the 3 points.

the resistivity by bout one order in magnitude.

We have simulated this effect as the result of diffusion

0.1

0.2

0.3

0.4

0.5

0.6

0.7

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0







Figure 8. Effect of the main size of dopant particles on con-

centration of the dopant atoms at the point

2 2.5xL= =

μm

between the neighbor dopant particles (simulation, see Fig-

ures 2 and 7). Content of dopant in DSG is 2 times less for

curve a than for curve b. The diffusion duration accords to

the diffusion lengthD

=1,25 μm.

process (Figure 8) in accordance with the expression (5)

for two val ues of c

<. There we assume that

( )

12N xL

−

∝=

, the same assumption is made in fig.

7. One can see that the dependence

( )

,2Nrx L=

becomes stronger for smaller

, when

the main distance between particles of the dopant have

been decreased for constant C, and conductivity of the

DSG increases.

It must be noted here that the information on the solu-

bility of RuO2 in lead-sili cate glass is contr adictory: Pala-

nisamy et al. [18] estimate it less than 10-4 atomic %,

while Flachbart et al. [8] present the value about 7 atom-

ic %. We are inclined to admit the last value because of

the results of X-ray diffraction experiments, in which the

intensity of main reflexes of RuO2 relicts in DSG reduces

nearly 10% due to firing and they become wider [19].

One should also take into account the fact that solubility

of pure metals in silicate glass is very small (really less

than 10-3 atomic%, [20]), while solubility of their oxides

is essentially higher (up to 10 – 70 mol . % for PbO).

It is clear in the same approach that the electrical

properties of DSG, specifically

( )

, beyond the forma-

tion of infinite cluster from diffusion zones, mainly de-

pend on the properties of the doped glass, namely, on the

glass structure, on the distribution of the energy levels of

the impurity in the energy gap of the glass, and on other

microscopic characteristics.

The resistivity of the glass in the diffusion zone is 5 - 6

orders of magnitudes higher than that of the dopant re-

licts, so the latter one does not affect the macroscopic

parameters of the DSG unless they form an infinite clus-

ter (i.e. unless c

<). Indeed, th e conduction is metal-

lic at c

3. Conclusions

The experiments and simulations show that the diffusion

zones are formed around the dopant particles in DSG due

On the Conduction Mechanism of Silicate Glass Doped by Oxide Compounds of Ruthenium (Thick Film Resistors)

to the diffusion of dopant atoms into the glass during the

firing. The glass becomes conductive in these zones and

the overlapping of them generates the infinite conductive

cluster (s) (percolation levels). This model allows ex-

plaining the effect of technological parameters such as

temperature and duration of firing, glass and dopant

composition, dispersivity of the powders, on the electric-

al properties of the DSG as well as the absence of perco-

lation threshold or its shift to smaller values of the do-

pant content

4. Acknowledgements

Fond for Support of Fundamental Researches of the Uz-

bek Academy of Sciences is acknowledged for the finan-

cial support (grants 55-08 and 27-10).

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