_{1}

^{*}

This article is the final part of the investigation of conduction mechanism of silicate glass doped by oxide compounds of ruthenium (thick film resistors). In the first part
[1]
, the formation of percolation levels due to diffusion of dopant atoms into the glass has been considered. The diffusion mechanism allowed us to explain shifting of the percolation threshold towards to lower value and the effect of firing conditions as well as the components composition on the electrical conduction of the doped glass. The coexistence of thermal activation and localization of free charge carriers as the result of nanocrystalline structure of the glass was the subject of the second part
[2]
. Because of it, the resistivity of the doped silicate glass is proportional to exp (–aT^{–}
^{ζ}
) at low temperatures (T <
50 K), 0.4 <
ζ
< 0.8. Structural transitions of nanocrystals take place at high temperatures (T > 800 K) and the conductivity of the doped silicate glass decreases sharply. We consider the origin of the minimum in the temperature dependence of resistivity of the doped silicate glass here. It is shown that the minimum arises from merge of impurity band into the valence band of glass at temperature high enough, so thermal activation of charge carriers as well as its hopping are failed, and scattering of free charge carriers become predominant factor in the temperature dependence of the resistivity.

This article is the final part of the investigation of conduction mechanism of silicate glass doped by oxide com- pounds of ruthenium (thick film resistors).

In the first part [^{?ζ}) at low temperatures (T < 50 K), 0.4 < ζ < 0.8. Struc- tural transitions of nanocrystals take place at high temperatures (T > 800 K) and the conductivity of the doped silicate glass decreases sharply.

An “enigmatic” minimum near the room temperature [

The manufacturing process of DSG used here is standard for technology of thick film resistors and was de- scribed elsewhere (the mixture of the glass and the dopant powders on the alumina substrate have been fired at T_{f} = 1073 - 1125 K in τ = 10 min). Content of RuO_{2} in our samples of DSG is indicated in the figure captions. The glass compositions investigated are as follows (weight %):

Glass1 SiO_{2} 27; PbO 67; BaO 4; MgO 2;

Glass2 SiO_{2} 32; PbO 63; Al_{2}O_{3} 5;

Glass3 SiO_{2} 33; PbO 67.

Firing temperature T_{f} is 1073 K for glass1, 1123 K for glass2 and glass3.

The value of resistance was measured by digital multimeter Sch-300 having error less than 0.2%. The tem- perature was measured by the same multimeter and standard Pt-PtRh thermocouple. The thermocouple was calibrated in standard temperature points of water boiling (373 K), tin and silver solidification (505 and 1234 K accordingly).

The DSG samples are screen-printed and have dimensions 10 × 10 × 0.025 mm^{3} so measured values of the re- sistance R (Ohm) and the resistivity ρ (Ohm・cm) are related as ρ = Rt = 2.5 × 10^{?3} R, here t = 2.5 × 10^{? 3 cm is the thickness of the DSG. }

The effective medium theory [^{*} and conductivity ρ^{*} of the DSG on the basis of known values of same parameters for the dopant and the glass (ε_{d}, ρ_{d} and ε_{g}, ρ_{g} accordingly). It is assumed in this theory that the volume content C of the dopant is known a priory. But experiments show that this assumption is not justified for the DSG due to physical or chemical interactions of components at the firing temperatures. These processes can generate new compounds or regions of unknown parameters. On the other hand, the minimum of ρ(T) can arise only from coexisting of at least the two compo- nents having opposite temperature dependence of conductivity―dielectric and metallic with same values of con- ductivity. Unfortunately these values are strong differing in the DSG: ρ_{g} > 10^{15} Ohm・cm and ρ_{d} < 2.5 × 10^{?4} Ohm・cm (for RuO_{2}, [

The percolation theory leads to the percolation threshold on the conductivity

here C_{c} is the critical volume content of the dopant (about 16 vol% in the three dimensions), t is the critical ex- ponent having most probable value about 1.7 in the three dimensions (must have the universality for various percolation problems in the space of same dimensions). σ_{0} is some constant.

The value of the critical exponent t for DSG is essentially differ [_{0} = 2 in percolation theory. This distinction has been explained as the result of tunneling and percolation [

Conductivity of DSG has been simulated by the Monte-Carlo method [

The tunneling-percolation mechanism [

here first factor in brackets is the temperature dependence of the tunneling through the barrier, the second one contains very small (about 0.4 - 5 meV) energy E, which is required to add into or remove the single charge car- rier from the dopant particle. ρ_{b} is the barrier height, ρ_{d0} is the resistivity of the dopant particle at T = 0 K, and k is the Boltzmann’s constant. Parameter a is the characteristic of the barrier height related to the Fermi level E_{F}, and researchers [

Expression (2) has a minimum, caused by metallic conductivity of the dopant particles (second term), but is applicable in the narrow range of temperature. On the other hand, the main distance between the dopant particles in the DSG

is about 0.1 - 1.5 μm. Appropriate thickness of the glass layer between the particles is L ? D ≈ 0.6 μm. It is clear that the tunneling of charge carriers is not effective on the same distances, so the description of DSG properties via tunneling runs against the problems. Here D is the diameter of the dopant particles, γ_{d} and γ_{g} are the specific weight of the dopant and the glass accordingly, C_{m} is the weight content of the dopant. L ≈ 0.87 μm for the powder of RuO_{2} with D ≈ 0.5 μm and C_{m} = 16 wt % (C ≈ 9.9%). The resonance tunneling has been used [

It should be noted also that sinaT/aT in (2) has the first minimum at aT ≈ 4.4 and the condition aT/3 < 1 is vi- olated. This minimum slightly shifts towards to lower values of aT due to the second term in (2) while experi- mental variation of T_{m} is 77 to 700 K in accordance with the glass composition, the doping level C and firing temperature T_{f}.

Fluctuation-induced tunneling conduction gives the temperature dependence [

here σ_{0}, T_{1} and T_{2} are some parameters. There is no maximum in (4) for T_{2} > 0 without including the metallic conductivity of dopant relicts and the percolation threshold should be introduced into the model via percolation theory.

Hopping-percolation model [

with_{wF} is the density of states of electrons near the Fermi level and n is the constant having most probable value about 1, A is some constant value. There is no maximum in (5) as well and hopping of carriers has not the threshold. Because of it hopping of carriers is combined with the percolation theory [

Low-temperature (T = 0.05 - 4.2 K) measurements show the hopping conduction of the DSG and the decreas- ing of the hopping energy due to generation of the narrow impurity band in the forbidden gap of the glass by diffusion of the ruthenium atoms. Concentration of the diffused atoms increases as T_{f} increases.

There are two problems. The first one caused by the possibility of experimental corroboration of the Mott’s low (5). It has been showed [

can give good agreement with the experimental data in the wide range of m = 0.2 - 0.55 if choose the value of n in the interval ?7.5 to 2.75. Despite it the expression (5) with various values of n is used often as the proof of the hopping conductivity in the DSG.

The second problem is the result of the fact that carriers hopping (which is actually the quantum-mechanical tunneling) is not effective in distance about 0.1 - 1.5 μm that corresponded to mean distance between the dopant particles in the DSG. It forces the researchers to consider the diffusion of dopant atoms into the glass or the par- ticles to be far smaller than they are actually.

Authors [

with x = 0.5 for DSG conductivity and explained it as result of tunneling though graded barriers. For x = 0.5 one finds p = 2 in three dimensions (d = 3), and this value corresponds to the density of states variations expected to result from Coulomb interaction between localized carriers [

by which the density of states g(E) rises about the Fermi level:

have calculated for mentioned experimental results the localization radius

charge, ε_{0} is the permittivity of free space, ε = 10 is the static dielectric constant of DSG. It was found that a_{0} ≈ 2 μm and the estimated activation energy less than kT. Such value of a_{0} is meaningless because of the mean size of the dopant particles where carriers to be localized is about 0.1 - 1 μm. So authors [

It was concluded in [

Conduction through the narrow impurity band generated in the forbidden band of the glass by diffused dopant atoms due to firing have been considered in [

Mismatch of the thermal expansion coefficients of the DSG and the substrate [

which takes into account the tension ψ of the DSG layer through the mismatch of thermal expansion coefficients of the glass and the ceramic substrate. Here ρ_{Θ} is the value of ρ(T) at temperature Θ, k_{Θ} is the gauge factor of DSG at Θ, χ is the temperature coefficient of the gauge factor. kβ is the activation energy of DSG.

This mismatch was taken for in [^{−6} K^{−1}.

Samples of the DSG glass + RuO_{2} without the ceramic substrate have been examined in [

Combination of the metallic and thermal activated (semiconducting) conductivity [

which is in the good agreement with experiment at T > T_{m}, but out of keeping at T < T_{m}. First term in (9) is the contribution of the glass layers between the dopant particles having metallic conductivity (the second one).

Prudenziati has pointed out [

Effects of firing temperature on the parameters of the DSG have been investigated in [

Unsoundness of the two-phase system model [

Electron microprobe analyses and atomic force microscope investigations [_{2} show that there are the zone of higher concentration of Ru atoms in the glass round the RuO_{2} par- ticles. These zones are formed by diffusion of Ru atoms into the glass. It is showed as well [_{2} based DSG is realized by hopping of carriers among the Ru clasters of 2 to 4 nm in size.

Unfortunately, relations of the diffusion zones and the minimum of the temperature dependence of DSG resis- tivity have not investigated in these works.

Let us to consider the DSG containing C_{m} = 16 wt% of RuO_{2} and 84 wt% of lead-silicate glass 2. C = 9.9% and the mean distance between centers of spherical particles of the dopant L = 0.87 μm for its diameter D = 0.5 μm here. DSG is the foil of l∙w∙t = 10×10×0.025 mm^{3}. Here w, l, t are width, length and thickness of the sample ac- cordingly. t = 0.0025 cm is the standard thickness of the thick film resistors.

Specific weights are γ_{d} = 6.85 g/cm^{3} for the dopant (RuO_{2}) and γ_{g} = 4 g/cm^{3} for the glass. Resistivity is ρ_{d} = 4 × 10^{?5} Ohm・cm (RuO_{2}) and ρ_{g} > 10^{16} Ohm・cm (the glass). Volume of the sample is v = 2.5 × 10^{?3} cm^{3}, volume of the dopant is v_{d} = Cv = 0.099 × 2.5 × 10^{−3} ≈ 2.5 × 10^{?4} cm^{3}.

We will estimate now the upper and lower limits of the resistivity for the DSG sample. There are two extreme distributions of the dopant particles in the sample (figure 1). Resistivity of the sample will be intermediate for all other cases.

In the first case the dopant forms the solid film of length l and thickness t along the electric field, i.e. the film closes the metallic terminations (figure 1(a)). Wideness of the same film is w_{d} = Cw = 0.099 cm and its resis-

tance is

parallel is

sample

In the second case the dopant film is across the electric field (figure 1(b)) and its resistance is

the temperature dependence of the sample resistance is determined by the dopant layer in the first case and by the glass layer in the second case:

and

Here R_{0} and R_{20} are values of resistance at some reference temperature T_{0}, ΔT = T ? T_{0} is the deviation of temperature from T_{0}. The minimum of the

is possible only if

near the minimum.

Simulation of the temperature dependence of the DSG resistance in these cases is shown in figure 2. In the case of the dopant layer along the electric field, R(T) is linear at low temperatures and has the maximum instead of the experimental minimum so this case is not applicable for DSG.

For this reason we will consider now the metallic and semiconducting sections connected in series (figure 1(b)) and attempt to find out the correlation between the resistivity values of these sections at the minimum of the to- tal resistivity. For this purpose we rewrite (9):

here a and 1 − a are the contributions of the semiconducting and metallic sections to the total resistivity. The condition of the minimum dρ(T)/dT = 0 at T = T_{m} gives a as a function of T_{m}:

a(T_{m}) as function of T_{m} is showed in figure 3. It is seen from figure 3 that a ≈ 0.3 - 0.9 is required to have T_{m} ≈

300 K, as it has often observed experimentally, and ρ(T_{m}) must be in order of the

well as_{d}(300 K) ≈ 4 × 10^{?5} Ohm・cm

for RuO_{2} and ρ_{g}(300 K) > 10^{16} Ohm・cm for the glass 2SiO_{2}・PbO while ρ(300 K) ≈ 10^{?2} - 10^{4} Ohm・cm for DSG. It means that the resistivity of RuO_{2} or other dopant particles cannot be used for interpret ρ(T_{m}) in terms of the (12) and the physical or chemical mechanism for reducing of the glass resistivity down to ρ(300 K) should be considered here.

The model and simulations It was pointed out above that the resistivity of the RuO_{2} used in DSG is lower of the resistivity of the DSG by factor of 10^{−3} - 10^{−4}. As a result the mechanism for lowering the ρ_{g} from 10^{16} Ohm∙cm down to the ρ of DSG (at least in order of value) should be considered.

Any of the tunneling of free carriers though the glass interlayer or hopping of them from the one dopant par- ticle to another cannot reduce the glass resistivity down to resistivity of the DSG as well as provide the R(T) with the minimum.

Therefore observed properties of the DSG can be caused by doping of the glass interlayer between the dopant particles due to diffusion of the dopant atoms only, and the temperature dependence ρ(T) ~ T or ρ(T) ~ T^{2} should be the property of the doped glass but no that of the RuO_{2} particles. This condition is satisfied for the electron- phonon or the electron-electron scattering.

So let us to consider now the combination of thermal activation, hopping and scattering of charge carriers as the main conduction mechanism of the DSG.

As reported earlier [

if the temperature dependence of the energy gap between the impurity band and the valence band of the glass is taken into account.

It is should be noted that the temperature dependence (13) with ς = 0.5 takes place in the hopping model of the doped semiconductors as the result of the Coulomb gap in the energy spectrum of the electrons as well [

We have taken into account therein the temperature dependence of width E_{G}(T) of the band-gap between the top of the glass valence band and the impurity band bottom caused by electron-phonon coupling [

and the temperature coefficient ς is about 10^{−6} eV・K^{−2}, so this effect is not essential in the wide-band semicon- ductors such as Ge, Si, GaAs or C (diamond). Here E_{G}(0) is the width of the band gap at T = 0 K.

Expression (14) is the experimental and is not suitable enough at low temperatures (T < Θ_{D}, here Θ_{D} is the Debye temperature).

The expression [

is more correctly at_{G}(0) is the band-gap width at T = 0 K (figure 4), ξ is the electron-pho-

non coupling constant, _{G} and it disappears at the certain temperature T_{m} in the semiconductors having the narrow band gap (less than 10 - 50 meV [

The impurity band and the valence band merge and form unite partially filled valence band as in the metals (figure 4) at T ≥ T_{m}, so concentration of the charge carriers become constant. The main factor of the tempera- ture dependence of resistivity of the DSG is thereafter the charge carriers scattering on phonons, neutral and io- nized impurities or on the other charge carriers, which lead to

or

Geometry of the impurity band in the glass band-gap at some characteristic temperatures is shown in figure 4. Temperature dependence of the impurity band-gap E_{G} is shown in figure 5.

Thermal activation and hopping of the charge carriers act simultaneously and increase the DSG conductivity while electron-phonon or electron-electron scattering decreases it so one can write for resistivity of DSG

here

is the conductivity due to thermal activation of carriers,

is the hopping conductivity, and

is the metallic conductivity, caused by scattering of carriers on neutral defects (ρ_{m0}), phonons (αT), charged im- purities (γT^{−3/2}) or other charge carriers (βT^{2}) [

It should be emphasized that

Experimental results Comparison of the experimental results of the temperature dependence of the resis-

tance R(T) of the DSG samples and our model (21) in the temperature range 160 - 800 K shows the good agree- ment (figures 7-9).

It is seen from these figures that

1) R(T) of the all investigated DSG is not quadratic

in contrast with the common opinion [

2) The temperature T_{m} of the minimum is affected by DSG composition and firing conditions (T_{f} and τ);

3) The temperature T_{m} of the minimum increases with the resistivity ρ of the DSG so can be distorted (figure 9) by structure transitions of silicate nanocrystals in the DSG [

1) The known conduction mechanisms of the DSG cannot explain the origin of the minimum of the tempera- ture dependence of resistivity and the effect of the glass and dopant composition, firing conditions (T_{f} and τ) on the its location (temperature T_{m} and resistivity ρ_{m}).

2) The minimum of the temperature dependence of the DSG resistivity is the result of decreasing of the ener- gy gap between the impurity band and the valence band of the glass by temperature. It is very important here to note that the impurity band-gap is narrow (about tens of meV) as the electron-phonon coupling constant has small value. So the band-gap can vanish at temperature T_{m} near the room temperature (the broad brand-gap will disappear at high T_{m} and we can’t observe its vanishing due to the structural transitions of nanocrystals [^{2} as in the typical metals.

Fund for Support of Fundamental Researches of the Uzbek Academy of Sciences is acknowledged for the finan- cial support (grants 27-10 and 14-12).